9th Class Math Full Book PDF

version: 1.1 CHAPTER Matrices and 1 Determinants Animation 1.1 : Matrix Source & Credit : eLearn.punjab 1. Matrices and Determinants eLearn.Punjab 1. Matrices and Determinants...

version: 1.1 CHAPTER Matrices and 1 Determinants Animation 1.1 : Matrix Source & Credit : eLearn.punjab 1. Matrices and Determinants eLearn.Punjab 1. Matrices and Determinants eLearn.Punjab Students Learning Outcomes 22. Verify the result (AB)-1 = B-1A-1 23. Solve a system of two linear equations and related real life After studying this unit , the students will be able to: problems in two unknowns using 1. Define Matrix inversion method, a matrix with real entries and relate its rectangular layout (formation) Cramer’ s rule. with real life, rows and columns of a matrix, Introduction the order of a matrix, equality of two matrices. The matrices and determinants are used in the field of Mathematics, 2. Define and identify row matrix, column matrix, rectangular matrix, Physics, Statistics, Electronics and other branches of science. The square matrix, zero/null matrix, diagonal matrix, scalar matrix, matrices have played a very important role in this age of Computer identity matrix, transpose of a matrix, symmetric and skew- Science. symmetric matrices. The idea of matrices was given by Arthur Cayley, an English 3. Know whether the given matrices are suitable for addition/ mathematician of nineteenth century, who first developed, “Theory subtraction. of Matrices” in 1858. 4. Add and subtract matrices. 5. Multiply a matrix by a real number. 1.1 Matrix 6. Verify commutative and associative laws under addition. 7. Define additive identity of a matrix. A rectangular array or a formation of a collection of real numbers, 8. Find additive inverse of a matrix. 1 3 4 say 0, 1, 2, 3, 4 and 7,such as, and then enclosed by 9. Know whether the given matrices are suitable for multiplication. 7 2 0 10. Multiply two (or three) matrices. 11. Verify associative law under multiplication. brackets `[ ]’ is said to form a matrix Similarly 12. Verify distributive laws. 13. Show with the help of an example that commutative law under is another matrix. multiplication does not hold in general (i.e., AB ≠ BA). We term the real numbers used in the formation of a matrix 14. Define multiplicative identity of a matrix. as entries or elements of the matrix. (Plural of matrix is matrices) 15. Verify the result (AB)t = BtAt. The matrices are denoted conventionally by the capital letters 16. Define the determinant of a square matrix. A, B, C, M, N etc, of the English alphabets. 17. Evaluate determinant of a matrix. 18. Define singular and non-singular matrices. 1.1.1 Rows and Columns of a Matrix 19. Define adjoint of a matrix. 20. Find multiplicative inverse of a non-singular matrix A and verify It is important to understand an entity of a matrix with the that AA-1 = I = A-1A where I is the identity matrix. following formation 21. Use adjoint method to calculate inverse of a non-singular matrix. 2 3 1. Matrices and Determinants eLearn.Punjab 1. Matrices and Determinants eLearn.Punjab 1 2 0 R1 In matrix A, the entries presented in horizontal (i) are equal matrices. way are called rows. A= 3 5 4 R2 In matrix A, there are three rows as shown We see that: 2 1 -1 R3 by R1, R2 and R3 of the matrix A. (a) the order of matrix A = the order of matrix B (b) their corresponding elements are equal. Thus A = B 2 3 5 In matrix B, all the entries presented in vertical way are called columns of the (ii) are not equal matrices. B= 0 1 -1 matrix B. We see that order of L = order of M but entries in the second row and 3 2 1 In matrix B, there are three columns as second column are not same, so L ≠ M. C1 C2 C3 shown by C1, C2 and C3. (iii) are not equal It is interesting to note that all rows have same number of elements and all columns have same number of elements but number matrices. We see that order of P ≠ order of Q, so P ≠ Q. of elements in rows and columns may not be same. EXERCISE 1.1 1. Find the order of the following matrices. 1.1.2 Order of a Matrix The number of rows and columns in a matrix specifies its order. If a matrix M has m rows and n columns, then M is said to be of order m-by-n. For example, 1 2 3  M= 1 0 2 is of order 2-by-3, since it has two rows and three    1 2 3 columns, whereas the matrix N = - 1 1 0  is a 3-by-3 matrix and 2. Which of the following matrices are equal?    2 3 7  P = [ 3 2 5 ] is a matrix of order 1-by-3. 1.1.3 Equal Matrices Let A and B be two matrices. Then A is said to be equal to B, and denoted by A = B, if and only if; (i) the order of A = the order of B (ii) their corresponding entries are equal. 3. Find the values of a, b, c and d which satisfy the matrix equation Examples 4 5 1. Matrices and Determinants eLearn.Punjab 1. Matrices and Determinants eLearn.Punjab (v) Null or Zero Matrix A matrix is called a null or zero matrix, if each of its entries is 0. 1.2 Types of Matrices 0 0 0  0 0  0  0 0 0  0 0 0  e.g., 0 0 [0 0] ,   0 0 0  and   (i) Row Matrix  , 0  ,  , 0 0 0 A matrix is called a row matrix, if it has only one row. are null matrices of orders 2-by-2, 1-by-2, 2-by-1, 2-by-3 and 3-by-3 e.g., the matrix M = [2 –1 7] is a row matrix of order 1-by-3 and respectively. Note that null matrix is represented by O. M = [1 –1] is a row matrix of order 1-by-2. (vi) Transpose of a Matrix (ii) Column Matrix A matrix obtained by interchanging the rows into columns or A matrix is called a column matrix, if it has only one column. columns into rows of a matrix is called transpose of that matrix. If A is  2 1 0  a matrix, then its transpose is denoted by At. e.g., M = 0 and N =   are column matrices of order 2-by-1   1  and 3-by-1 respectively.  1 2 3 1 2 - 1    2 1 4  e.g., (i) If A =  2 1 0  then At =   (iii) Rectangular Matrix  -1 4 -2  3 0 - 2 , A matrix M is called rectangular, if the number of rows of M is not equal to the number of M columns. 1 2 1 0 2 0 (ii) If B =  2 -1 3  then Bt =  1 1 2  7      a b c 8   2 3  e.g.,A = 1 1 B =  d e f  ; C = [1 2 3] and D =    2 3 ; 0 t are all rectangular matrices. The order of A is 3-by-2, the order of (iii) If C= [0 1 ], then C = B is 2-by-3, the order of C is 1-by-3 and order of D is 3-by-1, which indicates that in each matrix the number of rows ≠ the number of If a matrix A is of order 2-by-3, then order of its transpose At is 3-by-2. columns. (vii) Negative of a Matrix (iv) Square Matrix Let A be a matrix. Then its negative, -A is obtained by changing A matrix is called a square matrix, if its number of rows is equal the signs of all the entries of A, i.e., to its number of columns.  1 2 3  2 -1  - 1 0 - 2 e.g., A =  0 3  B=   and C =  ,  0 1 3  (viii) Symmetric Matrix A square matrix is symmetric if it is equal to its transpose i.e., are square matrices of orders, 2-by-2, 3-by-3 and 1-by-1 respectively. matrix A is symmetric, if At = A. 6 7 1. Matrices and Determinants eLearn.Punjab 1. Matrices and Determinants eLearn.Punjab = G k 0 0 For example 0 k 0 where k is a constant ≠ 0,1. 1 2 3 0 0 k  2 -1 4  e.g., (i) If M=   is a square matrix, then 2 0 0  3 4 0  3 0 Also A = 0 2 0 B=   and C = are scalar matrices of 0 0 2 0 3  1 2 3  2 -1 4  Mt =   = M. Thus M is a symmetric matrix. order 3-by-3, 2-by-2 and 1-by-1 respectively.  3 4 0   2 1 3  2 -1 3 (xii) Identity Matrix  -1 2 2  t 1 2 1 ≠ A A diagonal matrix is called identity (unit) matrix, if all diagonal (ii) If A=   then A =   3 1 3  ,  3 2 3 , entries are 1. It is denoted by I. Hence A is not a symmetric matrix. 1 0 0   1 0 e.g., A = 0 1 0 is a 3-by-3 identity matrix, B =   is a 2-by-2 0 0 1 0 1  (ix) Skew-Symmetric Matrix A square matrix A is said to be skew-symmetric, if At = –A. identity matrix, and C = is a 1-by-1 identity matrix.  0 2 3  -2 0 1  Note: (i) A scalar and identity matrix are diagonal matrices. e.g., if A=    -3 -1 0  (ii) A diagonal matrix is not a scalar or identity matrix.  0 -2 -3 = G 0 -2 -3 then A =t  2 0 -1 = -(-2) 0 -1 = = -A EXERCISE 1.2    3 1 0  -(-3) -(-1) 0 1. From the following matrices, identify unit matrices, row matrices, Since At = –A, therefore A is a skew-symmetric matrix. column matrices and null matrices. (x) Diagonal Matrix A square matrix A is called a diagonal matrix if atleast any one of the entries of its diagonal is not zero and non-diagonal entries are zero. 1 0 0 1 0 0 0 0 0  0 2 0    0 1 0  e.g., A=   B = 0 2 0  and C =   are all 2. From the following matrices, identify 0 0 3 0 0 2 0 0 3 , (a) Square matrices (b) Rectangular matrices diagonal matrices of order 3-by-3. (c) Row matrices (d) Column matrices  2 0 1 0 (e) Identity matrices (f) Null matrices M =  0 3 and N = 0 4 are diagonal matrices of order 2-by-2.     (xi) Scalar Matrix 3 1 2   -8 7 6 -4  1 0  (ii) 0  3 4 A diagonal matrix is called a scalar matrix, if 2 (i)   (iii)  (iv)  (v) all the diagonal entries are same and non-zero. 12 0 4   3 -2   0 1    1   5 6  8 9 1. Matrices and Determinants eLearn.Punjab 1. Matrices and Determinants eLearn.Punjab Addition of A and B, written A + B is obtained by adding the entries of the matrix A to the corresponding entries of the matrix B. 1  1 2 3 0 0  (vi) [3 10 -1] (vii) 0  (viii)  -1 2 0 (ix) 0 0        0   0 0 1  0 0  3. From the following matrices, identify diagonal, scalar and unit (identity) matrices. 1.3.2 Subtraction of Matrices If A and B are two matrices of same order, then subtraction of matrix B from matrix A is obtained by subtracting the entries of matrix B from the corresponding entries of matrix A and it is denoted by A – B. 4. Find negative of matrices A, B, C, D and E when: are conformable for subtraction. 5. Find the transpose of each of the following matrices: Some solved examples regarding addition and subtraction are given below. 6. Verify that if then (i) (At)t = A (ii) (Bt)t = B 1.3 Addition and Subtraction of Matrices 1.3.1 Addition of Matrices Let A and B be any two matrices. The matrices A and B are conformable for addition, if they have the same order. 2 3 0  -2 3 4 =e.g., A =  and B 1 are conformable for addition 1 0 6   2 3  10 11 1. Matrices and Determinants eLearn.Punjab 1. Matrices and Determinants eLearn.Punjab 1.3.4 Commutative and Associative Laws of Addition of Matrices (a) Commutative Law under Addition If A and B are two matrices of the same order, then A + B = B + A is called commulative law under addition. Note that the order of a matrix is unchanged under the operation of matrix addition and matrix subtraction. 1.3.3 Multiplication of a Matrix by a Real Number Let A be any matrix and the real number k be a scalar. Then the Thus the commutative law of addition of matrices is verified: scalar multiplication of matrix A with k is obtained by multiplying each A+B = B+A entry of matrix A with k. It is denoted by kA. (b) Associative Law under Addition  1 -1 4    If A, B and C are three matrices of same order, then Let A =  2 -1 0  be a matrix of order 3-by-3 and k = −2 be a real  -1 3 2  (A + B) + C = A + (B + C) is called associative law under addition. number. Then,  1 -1 4   (-2)(1) (-2)(-1) (-2)(4)  KA =- ( 2)  2 -1 ( 2) A =- 0  = (-2)(2) (-2)(-1) (-2)(0)       -1 3 2  (-2)(-1) (-2)(3) (-2)(2)   -2 2 -8 =  -4 2 0    -4   2 -6 Scalar multiplication of a matrix leaves the order of the matrix unchanged. 12 13 1. Matrices and Determinants eLearn.Punjab 1. Matrices and Determinants eLearn.Punjab is additive inverse of A. It can be verified as Thus the associative law of addition is verified: (A + B) + C = A + (B + C) 1.3.5 Additive Identity of a Matrix If A and B are two matrices of same order and A + B = A = B + A, then matrix B is called additive identity of matrix A. Since A + B = O = B + A. For any matrix A and zero matrix O of same order, O is called additive Therefore, A and B are additive inverses of each other. identity of A as A+O=A=O+A EXERCISE 1.3 1. Which of the following matrices are conformable for addition? 1.3.6 Additive Inverse of a Matrix If A and B are two matrices of same order such that 2. Find additive inverse of the following matrices: A+B=O=B+A, then A and B are called additive inverses of each other. Additive inverse of any matrix A is obtained by changing to negative of the symbols (entries) of each non zero entry of A. 3. If then find, 14 15 1. Matrices and Determinants eLearn.Punjab 1. Matrices and Determinants eLearn.Punjab (i) (A + B)t = At + Bt (ii) (A – B)t=At – Bt (i) (ii) (iii) c=+[-2 1 3 ] (iii) A + At is symmetric (iv) A – At is skew symmetric (v) B + Bt is symmetric (vi) B – Bt is skew symmetric (iv) (v) 2A (vi) (−1)B 1.4 Multiplication of Matrices (vii) (−2) C (viii) 3D (ix) 3C Two matrices A and B are conformable for multiplication, giving 4. Perform the indicated operations and simplify the following: product AB, if the number of columns of A is equal to the number of (i) (ii) rows of B. Here number of columns  1 2 3  1 1 1   1 -1 -1 +  2 2 2  (iii) [2 3 1] + ( [1 0 2] − [2 2 2] ) (iv) =-     of A is equal to the number of rows of B. So A and B matrices are  0 1 2   3 3 3  conformable for multiplication. Multiplication of two matrices is explained by the following examples. 2 0 2 0 (v) (vi) (i) If A = [1 2] and B =  3 1  then AB =   [1 2]  3 1  = [1 × 2 + 2 × 3 1 × 0 + 2 × 1] = [2 + 6 0 + 2] = [8 2], is a 1-by- 1 2 3 1 -1 1   -1 0 0  2 matrix. 2 3 1  B =  2 -2 2  and C =   0 -2 3  5. For the matrices A =    1 -1 0   3 1 3   1 1 2  (ii) If A = and B = then verify the following rules. (i) A + C = C + A (ii) A+B=B+A (iii) B + C = C + B (iv) A + (B + A) = 2A + B (v) (C − B) + A = C + (A − B) (vi) 2A + B = A + (A + B) (vii) (C−B) A = (C − A) − B (viii) (A + B) + C = A + (B + C) (ix) A + (B − C) = (A − C) + B (x) 2A + 2B = 2(A + B) 1.4.1 Associative Law under Multiplication 6. find (i) 3A − 2B If A, B and C are three matrices conformable for multiplication then associative law under multiplication is given as (AB)C = A(BC) (ii) 2At − 3Bt.  2 3  0 1  2 2 7. then find a and b. e.g., A =  -1 0  B =  3 1 and C =  -1 0  then       8. then verify that L.H.S. = (AB)C 16 17 1. Matrices and Determinants eLearn.Punjab 1. Matrices and Determinants eLearn.Punjab R.H.S. = AB + AC  2 3  0 1  2 2   R.H.S = A(BC) =  -1 0   3 1  -1 0       Which shows that A(B + C) = AB + AC; Similarly we can verify (ii). (b) Similarly the distributive laws of multiplication over subtraction are as follow. The associative law under multiplication of matrices is verified. (i) A(B - C) = AB -AC (ii) (A - B)C = AC - BC 1.4.2 Distributive Laws of Multiplication over Addition and Subtraction (a) Let A, B and C be three matrices. Then distributive laws of multiplication over addition are given below: (i) A(B + C) = AB + AC (Left distributive law) (ii) (A + B)C = AC + BC (Right distributive law) L.H.S = A (B+C) R.H.S. = AB − AC 18 19 1. Matrices and Determinants eLearn.Punjab 1. Matrices and Determinants eLearn.Punjab 1.4.4 Multiplicative Identity of a Matrix Let A be a matrix. Another matrix B is called the identity matrix of A under multiplication if which shows that AB = A = BA A(B – C) = AB – AC; Similarly (ii) can be verified. 1.4.3 Commutative Law of Multiplication of Matrices  0 1 1 0 Consider the matrices A=   and B=   , then 2 3 0 -2   0 1 1 0   0 ×1 + 1× 0 0 × 0 + 1(-2)   0 -2  AB=  =    =     2 3 0 -2   2 ×1 + 3 × 0 2 × 0 + 3(-2)   2 -6  1 0   0 1   1× 0 + 0 × 2 1× 1 + 0 × 3   0 1 and BA =  =    =    0 -2   2 3 0 × 0 + (-2) × 2 0 × 1 + 3(-2)   -4 -6  = =0 -3G 1 2 Which shows that, AB ≠ BA Which shows that AB = A = BA. Commutative law under multiplication in matrices does not hold in 1.4.5 Verification of (AB)t = Bt At general i.e., if A and B are two matrices, then AB ≠ BA. Commutative law under multiplication holds in particular case. If A, B are two matrices and At, Bt are their respective  2 0  -3 0  transpose, then (AB)t = BtAt. e.g., if A = 0 1 and B =  0 4  then     Which shows that AB = BA. 20 21 1. Matrices and Determinants eLearn.Punjab 1. Matrices and Determinants eLearn.Punjab 5. verify t t t whether Thus (AB) = B A (i) AB = BA. (ii) A(BC) = (AB)C EXERCISE 1.4 (iii) A(B + C) = AB + AC (iv) A(B − C) = AB − AC 6. For the matrices 1. Which of the following product of matrices is conformable for multiplication? Verify that (i) (AB)t = Bt At (ii) (BC)t = Ct Bt. 1.5 Multiplicative Inverse of a Matrix 1.5.1 Determinant of a 2-by-2 Matrix be a 2-by-2 square matrix. The determinant of A, 3 0 6 2. If A = =  , B 5  , find (i) AB (ii) BA (if possible)  -1 2    denoted by det A or A is defined as 3. Find the following products. 4. Multiply the following matrices. 2 3 1 2 1.5.2 Singular and Non-Singular Matrix 2 -1 1 2 3  (a)  1 1  3 4 0  (b)     3  4 5 6     0 -2   -1 1  A square matrix A is called singular, if the determinant of A is 1 2  5 equal to zero. i.e., A= 0. 2 -  1 2 3  8 5   2 4    1 2   4 5 6  (c ) 3 (d )   For example, A = is a singular matrix,   6 4   -4 4  0 0   -1 1      since det A = 1 × 0 – 0 × 2 = 0  -1 2  0 0  (e)  A square matrix A is called non-singular, if the determinant of A is not 1 3  0 0  22 23 1. Matrices and Determinants eLearn.Punjab 1. Matrices and Determinants eLearn.Punjab = G 1 1  d -b  Adj M equal to zero. i.e., A ≠ 0. For example, A = 0 2 is non-singular, and Adj = M   , then = M-1  -c a  M since det A = 1 × 2 – 0 × 1 = 2 ≠ 0.Note that, each square matrix with real entries is either singular or non-singular. 2 1 e.g., Let A=   , Then  - 1 -3  1.5.3 Adjoint of a Matrix 2 1 A= =-6 - (-1) =-6 + 1 =-5 ≠ 0 -1 -3 Adjoint of a square matrix A = is obtained by  -3 -1 3 1    5 Adj A  1 2  -1  -3 -1 5 Thus = A -1 = = =   interchanging the diagonal entries and changing the signs of other A -5 5  1 2   -1 -2  entries. Adjoint of matrix A is denoted as Adj A.  5 5  3 1   6 1 2 2   - - 2 1 5 5  5 5 5 5  and AA =  -1   -1 -2  =  3 3   -1 -3   - + 1 6 - +  5 5   5 5 5 5  1 0  =  = I= AA -1 0 1  1.5.4 Multiplicative Inverse of a Non-singular Matrix 1.5.6 Verification of (AB)–1 = B–1 A–1 Let A and B be two non-singular square matrices of same order.  3 1 0 -1 Let A =  -1 0  and B =  3 2  Then A and B are said to be multiplicative inverse of each other if    AB = BA = I. Inverse of Identity Then det A = 3 × 0 – (–1) × l = 1 ≠ 0 The inverse of A is denoted by A-1 , thus matrix is Identity and det B = 0 × 2 – 3(–1) = 3 ≠ 0 AA–1 = A–1 A = I. matrix. Therefore, A and B are invertible i.e., their inverses exist. Inverse of a matrix is possible only if matrix is non-singular. Then, to verify the law of inverse of the product, take  3 1   0 -1  3 × 0 + 1 × 3 3 × (-1) + 1 × 2   3 -1 1.5.5 Inverse of a Matrix using Adjoint =AB =     =  -1 0   3 2   -1 × 0 + 0 × 3 -1 × (-1) + 0 × 2   0 1   3 -1 ⇒ det (AB) = = 0 1 =3≠0 be a square matrix. To find the inverse of  1 1 M, i.e., M-1, first we find the determinant as inverse is possible only 1 1 1  3 −1 and L.H.S. = (AB) =  0 3 =  3  of a non-singular matrix. 3  0 1 1  2 1 1 0 -1 R.H.S. = B−1A−1, where B−1 = A−1 = 3  -3 0  , 1 1 3 24 25 1. Matrices and Determinants eLearn.Punjab 1. Matrices and Determinants eLearn.Punjab 1  2 1  1 0 -1 1  2 × 0 + 1× 1 2 × (-1) + 1× 3  =  .   = 3  -3 0  1 1 3 3  -3 × 0 + 0 ×1 -3 × (-1) + 0 × 3  1 0 + 1 -2 + 3 1 1 1  1 1 6. then verify = =  = 3 3 3  0 3  3 0 3  0 1   that (i) (AB)−1 = B−1A−1 −1 −1 −1 = (AB)−1 Thus the law −1 (AB) = B A-1 −1 is verified. (ii) (DA) = A D EXERCISE 1.5 1.6 Solution of Simultaneous Linear Equations 1. Find the determinant of the following matrices. System of two linear equations in two variables in general form is given as ax + by = m cx + dy = n where a, b, c, d, m and n are real numbers. This system is also called simultaneous linear equations. 2. Find which of the following matrices are singular or non-singular? We discuss here the following methods of solution. (i) Matrix inversion method (ii) Cramer’s rule (i) Matrix Inversion Method 3. Find the multiplicative inverse (if it exists) of each. Consider the system of linear equations ax + by = m cx + dy = n 4. (i) A(Adj A) = (Adj A) A = (det A)I (ii) BB-1 = I = B−1B 5. Determine whether the given matrices are multiplicative inverses of each other. 26 27 1. Matrices and Determinants eLearn.Punjab 1. Matrices and Determinants eLearn.Punjab Example 1 Solve the following system by using matrix inversion method. 4x – 2y = 8 3x + y = –4 Solution 4 -2   x   8  Step 1 3 =  1   y   -4  4 -2  Step 2 The coefficient matrix M =  is non-singular, 3 1  since det M = 4 × 1− 3(−2) = 4 + 6 = 10 ≠ 0. So M–1 is possible. (ii) Cramer’s Rule  x -1  8  1  1 2  8  Consider the following system of linear equations. Step 3 = y M=  -4  10  -3 4   -4         ax + by = m 1  8-8  1  0   0  cx + dy = n = = = 10  -24 - 16  10  -40   -4  We know that  x  0  ⇒  y =      -4  ⇒ x=0 and y = -4 Example 2 Solve the following system of linear equations by using Cramer’s rule. 3x - 2y = 1 -2x + 3y = 2 Solution 3x - 2y = 1 -2x + 3y = 2 We have  3 -2   1 -2  3 1 =A = A  x 2 3 , = , A  -2 2   -2 3  y     3 -2 A = = 9 - 4 = 5 ≠ 0 (A is non-singular -2 3 28 29 1. Matrices and Determinants eLearn.Punjab 1. Matrices and Determinants eLearn.Punjab Thus, by the equality of matrices, width of the rectangle x = 19 cm 1 -2 and the length y = 51 cm. Ax 2 3 3+ 4 7 =x = = = Verification of the solution to be correct, i.e., A 5 5 5 p = 2 × 19 + 2 × 51 = 38 + 102= 140 cm 3 1 Also y = 3(19) – 6 = 57 – 6 = 51 cm Ay -2 2 6+2 8 =y = = = A 5 5 5 EXERCISE 1.6 Example 3 1 Use matrices, if possible, to solve the following systems of linear The length of a rectangle is 6 cm less than three times its width. equations by: The perimeter of the rectangle is 140 cm. Find the dimensions of the (i) the matrix inversion method (ii) the Cramer’s rule. rectangle. (by using matrix inversion method) 2x − 2y = 4 2x + y = 3 (i) (ii) 3x + 2y = 6 6x + 5y = 1 Solution 4x + 2y = 8 3x − 2y = −6 (iii) (iv) If width of the rectangle is x cm, then length of the rectangle is 3x − y = −1 5x − 2y = −10 y = 3x – 6, 3x − 2y = 4 4x + y = 9 (v) (vi) from the condition of the question. −6x + 4y = 7 −3x − y = −5 The perimeter = 2x + 2y = 140 (According to given condition) 2x − 2y = 4 3x − 4y = 4 (vii) (viii) ⇒ x + y = 70 ……(i) −5x − 2y = −10 x + 2y = 8 and 3x – y = 6 ……(ii) Solve the following word problems by using In the matrix form (i) matrix inversion method (ii) Crammer’s rule. 2 The length of a rectangle is 4 times its width. The perimeter of 1 1   x  70  the rectangle is 150 cm. Find the dimensions of the rectangle. 3 =  -1  y   6  3 Two sides of a rectangle differ by 3.5cm. Find the dimensions 1 1 1 1 of the rectangle if its perimeter is 67cm. det  = =1 × (-1) - 3 × 1 =-1 - 3 =-4 ≠ 0 3 -1 3 -1 4 The third angle of an isosceles triangle is 16° less than the sum of the two equal angles. Find three angles of the triangle. We know that 5 One acute angle of a right triangle is 12° more than twice the other acute angle. Find the acute angles of the right triangle. AdjA =X A= -1 B and A -1 6 Two cars that are 600 km apart are moving towards each A other. Their speeds differ by 6 km per hour and the cars are  x  1 -1 -1 70  Hence   =   123 km apart after hours. Find the speed of each car.  y  -4 -3 1  6   76  -1 -70 - 6 -1  -76   4  19  = = = =  4 -210 + 6 4  -204   204  51  4  30 31 1. Matrices and Determinants eLearn.Punjab 1. Matrices and Determinants eLearn.Punjab Review Exercise 1 SUMMARY 2. Complete the following: A rectangular array of real numbers enclosed with brackets is said = 0 0 G to form a matrix. (i) 0 0 is called..... matrix. A matrix A is called rectangular, if the number of rows and number = G 1 0 (ii) is called..... matrix. of columns of A are not equal. 0 1 A matrix A is called a square matrix, if the number of rows of A is = G 1 -2 (iii) Additive inverse of 0 -1 is.......... equal to the number of columns. (iv) In matrix multiplication,in general, AB...... BA. A matrix A is called a row matrix, if A has only one row. (v) Matrix A + B may be found if order of A and B is...... A matrix A is called a column matrix, if A has only one column. (vi) A matrix is called..... matrix if number of rows and columns A matrix A is called a null or zero matrix, if each of its entry is 0. are equal. Let A be a matrix. The matrix At is a new matrix which is called transpose of matrix A and is obtained by interchanging rows of A = = G, a+3 4 -3 4 3. If = 6 b-1 G 6 2 then find a and b. into its respective columns (or columns into respective rows). A square matrix A is called symmetric, if At = A. 2 3 4. If A = = 1 0G , B = =5 -4 G , then find the following. Let A be a matrix. Then its negative, −A, is obtained by changing the -2 -1 signs of all the entries of A. (i) 2A + 3B (ii) -3A + 2B A square matrix M is said to be skew symmetric, if Mt = −M, 2 (iii) -3(A + 2B) (iv) 3 (2A - 3B) A square matrix M is called a diagonal matrix, if atleast any one of entry of its diagonal is not zero and remaining entries are zero. 5. Find the value of X, if =2 1 G+ X = =4 -2 G. A diagonal matrix is called identity matrix, if all diagonal entries are 3 -3 -1 -2 If A = = 0 G, 4 B= =5 G 1 -3 6. -2 , then prove that 1 0 0  2 -3 1. A = 0 1 0  is called a 3-by-3 identity matrix. (i) AB ≠ BA (ii) A(BC) = (AB)C   0 0 1  7. If A = = 3 2 G and B = = 2 4 G, then verify that Any two matrices A and B are called equal, if 1 -1 -3 -5 (i) order of A= order of B (ii) corresponding entries are same (i) (AB)t = BtAt (ii) (AB)-1 = B-1 A-1 Any two matrices M and N are said to be conformable for addition, if order of M = order of N. Let A be a matrix of order 2-by-3. Then a matrix B of same order is said to be an additive identity of matrix A, if B+A=A=A+B 32 33 1. Matrices and Determinants eLearn.Punjab 1. Matrices and Determinants eLearn.Punjab Let A be a matrix. A matrix B is defined as an additive inverse of A, The solution of a linear system of equations, if B+A=O=A+B ax + by = m Let A be a matrix. Another matrix B is called the identity matrix of cx + dy = n A under multiplication, if B × A = A = A × B. a b   x  m by expressing in the matrix form  = c d   y   n  a b -1 Let M =  be a 2-by-2 matrix. A real number λ is called  x a b  m c d  is given by   =   y  c d   n  determinant of M, denoted by det M such that a b if the coefficient matrix is non-singular. det M = = ad − bc = λ By using the Cramer’s rule the determinental form of solution of c d equations A square matrix M is called singular, if the determinant of M is equal to zero. ax + by = m A square matrix M is called non-singular, if the determinant of M is cx + dy = n not equal to zero. is a b For a matrix M =  , adjoint of M is defined by c d  m b a m  d -b  Adj M =  . n d c n a b  - c a  x= and y= , where ≠0 a b a b c d a b c d c d Let M be a square matrix  , then c d  1  d -b  M−1 =  a  , where det M = ad − bc ≠ 0. ad - bc  -c The following laws of addition hold M+N=N+M (Commutative) (M + N) + T = M + (N + T) (Associative) The matrices M and N are conformable for multiplication to obtain MN if the number of columns of M = number of rows of N, where (i) (MN) ≠ (NM), in general (ii) (MN)T = M(NT) (Associative law) (iii) M(N + T) = MN + MT (iv) (N + T)M = NM + TM Law of transpose of product } (Distributive laws) (AB)t = Bt At (AB)−1 = B−1 A−1 AA−1 = I = A−1A 34 35  version: 1.1 CHAPTER REAL AND 2 COMPLEX NUMBERS Animation 2.1:Real And Complex numbers Source & Credit: eLearn.punjab 2. Real and Complex Numbers eLearn.Punjab 2. Real and Complex Numbers eLearn.Punjab Students Learning Outcomes 2.1 Real Numbers After studying this unit , the students will be able to: Recall the set of real numbers as a union of sets of rational and We recall the following sets before giving the concept of real numbers. irrational numbers. Depict real numbers on the number line. Natural Numbers Demonstrate a number with terminating and non-terminating The numbers 1, 2, 3,... which we use for counting certain objects recurring decimals on the number line. are called natural numbers or positive integers. The set of natural Give decimal representation of rational and irrational numbers. numbers is denoted by N. Know the properties of real numbers. i.e., N = {1,2,3,....} Explain the concept of radicals and radicands. Differentiate between radical form and exponential form of an Whole Numbers expression. If we include 0 in the set of natural numbers, the resulting set is Transform an expression given in radical form to an exponential the set of whole numbers, denoted by W, form and vice versa. i.e., W = {o,1,2,3,....} Recall base, exponent and value. Apply the laws of exponents to simplify expressions with real Integers exponents. The set of integers consist of positive integers, 0 and negative integers Define complex number z represented by an expression of the and is denoted by Z i.e., Z = {..., –3, –2, –1, 0, 1, 2, 3,... } form z= a + ib , where a and b are real numbers and i= -1 Recognize a as real part and b as imaginary part of z = a + ib. 2.1.1 Set of Real Numbers Define conjugate of a complex number. Know the condition for equality of complex numbers. First we recall about the set of rational and irrational numbers. Carry out basic operations (i.e., addition, subtraction, multiplication and division) on complex numbers. Rational Numbers All numbers of the form p/q where p, q are integers and q is Introduction not zero are called rational numbers. The set of rational numbers is denoted by Q, The numbers are the foundation of mathematics and we use  p  i.e=., Q  | p, q ∈ Z ∧ q ≠ 0  different kinds of numbers in our daily life. So it is necessary to be q  familiar with various kinds of numbers In this unit we shall discuss real Irrational Numbers numbers and complex numbers including their properties. There is a The numbers which cannot be expressed as quotient of integers one-one correspondence between real numbers and the points on the are called irrational numbers. real line. The basic operations of addition, subtraction, multiplication The set of irrational numbers is denoted by Q’,  p  and division on complex numbers will also be discussed in this unit. Q=′  x | x ≠ , p, q ∈ Z ∧ q ≠ 0  Version: 1.1  q  Version: 1.1 2 3 2. Real and Complex Numbers eLearn.Punjab 2. Real and Complex Numbers eLearn.Punjab The number ‘a’ associated with a point P on l is called the For example, the numbers 2, 3, 5, p and e are all irrational numbers. coordinate of P, and l is called the coordinate line or the real number The union of the set of rational numbers and irrational numbers is line. For any real number a, the point P’(– a) corresponding to –a lies known as the set of real numbers. It is denoted by R, at the same dist