Business Mathematics PDF Notes

Business Mathematics 1 Block – 1: Matrices and Determinants Notes...

Business Mathematics 1 Block – 1: Matrices and Determinants Notes e Course Contents: in Definition of a Matrix Types of Matrices Algebra of Matrices nl Properties of Determinants Adjoint of a Matrix O Inverse of Matrix The Rank of a Matrix Solution of Linear Equation System by Cramer’s rule ity Solution of Linear Equation System by Matrix Inverse Method Key Learning Objectives: At the end of this block, you will be able to: rs 1. Define matrix 2. Identify types of matrices 3. Describe algebra of matrices ve 4. Describe properties of determinant 5. Define adjoint and inverse of a matrix 6. Evaluate the solution system of linear equations by Cramer’s rule ni 7. Evaluate the solution system of linear equations by Matrix Inverse Method Structure: U Unit 1.1: Definition of Matrix 1.1.1 Introduction ity 1.1.2 Definition of Matrix 1.1.3 Types of Matrices 1.1.3.1 Row and Column Matrices m 1.1.3.2 Zero or Null Matrix 1.1.3.3 Square Matrix 1.1.3.4 Diagonal Matrix )A 1.1.3.5 Rectangular Matrix 1.1.3.6 Scaler Matrix 1.1.3.7 Symmetric Matrix (c 1.1.3.8 Skew-symmetric Matrix 1.1.3.9 Unit or Identity Matrix Amity Directorate of Distance & Online Education  2 Business Mathematics 1.1.3.10 Upper Triangular Matrix Notes e 1.1.3.11 Lower Triangular Matrix 1.1.4 Equal Matrix in Unit 1.2: Algebra of Matrices 1.2.1 Introduction nl 1.2.2 Algebra of Matrices 1.2.2.1 Addition of Matrices O 1.2.2.2 Difference of two Matrices 1.2.2.3 Multiplication of Matrices 1.2.2.4 Negative Matrix ity 1.2.3 Properties of Matrix Addition 1.2.4 Properties of Multiplication of Matrices 1.2.5 Transpose of a Matrix rs 1.2.6 Properties of Transpose of Matrices Unit 1.3: Determinants ve 1.3.1 Introduction 1.3.2 Determinants 1.3.3 Properties of Determinants ni Unit 1.4: Rank and Inverse of a Matrix 1.4.1 Introduction U 1.4.2 Minor and Cofactors 1.4.3 Adjoint of Matrix 1.4.4 Singular Matrix ity 1.4.5 Non-singular Matrix 1.4.6 Inverse Matrix 1.4.7 Rank of a Matrix m Unit 1.5: Linear Equation 1.5.1 Introduction )A 1.5.2 Linear equation 1.5.2.1 Solution of a system of Linear equations by Cramer’s rule 1.5.2.2 Solution of a system of Linear equations by Matrix Inverse Method (c Amity Directorate of Distance & Online Education  Business Mathematics 3 Unit - 1.1: Definition of Matrix Notes e Unit Outcome: in At the end of this unit, you will be able to: Define Matrix nl Define various types of Matrices Define Equal Matrix O 1.1.1 Introduction Knowledge of matrices is required in various branches of mathematics. Matrix is one of the most powerful tools of mathematics. Compared to other straightforward ity methods, this mathematical tool makes our work much easier. The concept of the matrix evolved as an attempt to solve the system of linear equations in a short and simple form. Matrix notation or representation and operations are used to create electronic spreadsheet programs for the personal computer that is used in various fields of commerce and science like budgeting, sales projection, cost estimation, analysis of the rs result of an experiment, etc. In this unit, we will discuss the matrix, equal matrix and its various types. ve 1.1.2 Definition of Matrix An arrangement of m x n numbers or functions in the form of m horizontal lines (called rows) and n vertical lines (called columns), is called a matrix of the type m by n (or m x n). ni Such an array is enclosed by the bracket [ ]. Each of the m, n numbers constituting the matrix is called an element of the matrix. U The location of each element in the matrix is fixed. Therefore, the elements of the matrix are represented by letters which have two subscripts. The first subscript row and the second subscript column reveal which row and which column the element is in. Thus, the element in the ith row and jth column is written with aij. Therefore, the matrix in ity the m row and n column is often written as follows  a11 a1n  Amn     m a amn   m1 This is called a matrix of m x n. )A 2the8number The first letter of the m x n represents 3  of rows in the matrix A and the second letter the number of its columns. 5 7 4     Hence, we can say: 2 5  2 8 3  8 7  (c 5 7 4  is a 2 x 3 matrix, and a 3 x 2 matrix.     3 4  2 5  Amity Directorate of Distance & Online Education 8 7  2  4      3 4   5   8 32 2 8 38 7  5 75 47 4  2 8 3  2 8 3       3 4 2 5  5 7 4  25 8 2 8 7 43   2 5   2   5 7 43  8 7    8 7   4  2 5  4 Business Mathematics  1.1.3 Types of Matrices 225 875  43  3 4     8 7  Notes 2 5   3 4   5    e 8 7 1.1.3.1 Row and 225Column75  4matrix- 8   3  2  A matrix having only  3 4  row is called a row one matrix while the one having 8 7  one 0 as0a column 0 83 47only column is known 4  2  matrix. in 5 7 4 23 45      4    0 0 2 0  Example 1.1.1 [2    2836 3]47 is a (1 x3)row  5    matrix while  4  is a (3 x 1) column matrix.  422 5  5       0 0 5 0  nl  23 4  0 0 0   0 0 0   8 4 5  7 0 each 0 of 0 whose 1.1.3.2 Null Matrix-  4  Anm x n0matrix 0 0    elements is 0, is called a null matrix of the type m x 3 5  4      0 0 0 0 0 0   25   n. 0 0 0      40  0 0  O 20  0 0   0 0 0  0 0 0    0  0 0  Example 1.1.2  0 5  0 0  ,  0 0 0 00 and are null matrices of the type  40  0 0    0 0 0  0 00 0 0   0 0 0 0 0 0   0 0 0  05  0 0   ity    0 0 0    2 x 3, 3 x 3 and 2 x 2, 0respectively. 0 0   A  aij0 nn0 0  0 0 0 0 00  0 0    1.1.3.3 Square Matrix- 0 0 An 0  mx n matrix  0 0for which m = n, i.e., the number of rows 0 0 0  0 0   1matrix 90 of25 equals the number of0 columns 0 0  is called0 a 0square  0order n or an n- rowed  0 0 0       square matrix. 0 0  0  4 12 30 0 0  The element aij of elements of the matrix  0 0  0  A  aij  nn 00a square A0and 0 0aij 0nn 4 12 0 rs 0 0  0 1 9 25  the  line along 1 30 which 8 16 32  00  0 matrix, A  aij  nn for which i = j, is called the diagonal they 9  25 1 lie is 0 A  aij  nn called 0  or simply the diagonal ve 4 12 30 0 21 0  9 25  of the matrix. A0 0aij  nn  A10 0a9ij  n25 8 16  32       n    8 16 32 0 04 3 12 30  Example 1.1.3 4 1 12 9 25  is a square matrix of order A1 0 0a9ij  30 1 0 0  8 16 3. 32   n25 n  1 0 0  1 2 3  ni 8 4 16 12 30 32   A4 12 30  0 2 0  2 314 0 0  which 1.1.3.4 Diagonal 8 1 16 Matrix- a9ij  n2532 A  0 2 0  each one of the non-diagonal n square matrix in  18 16 0 0 32   0 0 3   0 2 0  matrices is 0, is called4a diagonal 10 12 30 matrix.  0 0 3      U 209 025  2 0 0 0 0 3  18 16 0 032  1 2 3   0 2 0   40 12 20 0330   1 2 3   of order 3 x 3. Example 1.1.4 0 2 0 is 3 4  matrix  2a diagonal 180 16 0 0332   2 3 4   0 01 2 2 3  10 20 33   2 3 4  ity  0 2 0   2 0 0    1 1.3.1.5 Rectangular 2 230Matrix- 430 A m2x n 0matrix  1 in 3 which 4  the number of rows and  0 20 33 0 2 0 0   columns is not same,1 20 32 40  is called a rectangular  0 2 0  3 22 5 0 0  matrix.  2 40  0 0 2     0 30  4 5  0 6 2 0  1 20 33  0 0 2      Example 1.1.5 2 0 20 0 is a rectangular m matrix of order 2 x 3. 22 30 40  1 3 4   0 0 2  10 20 320  1 3 4  1.1.3.6 Scalar Matrix- 0 2 A0diagonal  3 2 matrix 5  in which all the diagonal elements are 2 3 40 0 0 2  3 2 5  1 3 4   0 03 42   4 5 6  )A equal is called a scaler 1 matrix. 3 2 5 0 213 203 405  2 0   4 5 6      4 5 6   1 0 03 42  3 2 5  Example 1.1.6  0   4 5 60  is a scaler matrix of order 3 x 3.  1 3 2 5 4 035 4 0 62   43 25 65  (c  1 3 4   4 5 6  3 2 5 Amity Directorate of Distance & Online Education  4 5 6    0 0 3  1 2 3  2 3 4    A'  A ' Business A A Mathematics 2 0 0  5 '  0 2 0 A  A ' A  aij to be symmetric if ' A  A A'  1.1.3.7A A  A i.e., if A  aij  Symmetric  ' Matrix- A square matrix is said n Notes e A'  A A A  n 0 0 2 A  a  A  aij  A  aij  then A is called symmetric  matrix ij  n iff a ij a ji  i & j. A  aij  n ' a  i& j n aij A  jiA A A1A' ' a  aij3Aij 4n n in A  An a a Ai' & A jA'   A a  a  i & j a ij' a ji  i & j   ij ji ij a  ji a  i &j A A Example Aaij  1.1.7  a 3ij ' 2aAajiij 5 A &  iis& ajsymmetric matrix of order 3 x 3. ij ji a a   i j    4 5 ij 6nA   A A  aAij  naij  n n ij A  ji a  n ' ' A   A A'   A A '  A nl A a aija  i & j ' AaA '  a    Aji  ij jin A   ij' Aa ij n i& j A  aij  A  1.1.3.8 aij  Skew–Symmetric a A a jiAMatrix Ai  &ja–ij Ana square  i& a ji0matrix x j y is said to be skew–symmetric n ij ij  A  anij  if 0A   ' x A yi.e.,  if A A aA '   ijaaijjiAthen  iA &isj called skew–symmetric   x 0 z  matrix. n 'A ij  a n  O   A  A ij  n0 n x Ay  A '  0 x y  0x x0 yz      0 x  y A  aij    xaA   yx 0 z    y  z 0    x 0z  z  x 0 '  0  A 0 x y   y  z n0  a     x 0 z   A ij   ij  aijajin  i& j n A  aij   xA x 0a0 zzy isaz skew–symmetric 0  1 n 0  matrix.   y  z 0     y Example.  z 0  1.1.8    y  z 0  1 00  x y   y0A0y'z xxzA  n 0y ij ity 0  y    0  x  y     x01  0 z     x 0 1 z 0   0 1  1 0  10 1.1.3.9  Unit or identity 0  x 0matrix- x z y  Scalar  xmatrix 0 z each of whose diagonal0 11element  0  is 0  y1   z 0  11A y  0 aijz 00 1     0  1  unity,  is called a unit matrix 0 0z an yx or  n0identity z  matrix.   y  1Itzis0represented 0 0  by ‘I’. 1 0 0   0 01  1     0 1 0   1 0      y  1 0 0  x1 y 0 0  z 0    1 0 0  rs 10 01 0 1 0    1 00 0 1   0 11 00  0  00Example 1 0 1    1 0 1 010 100  1.1.9  x 0 1 , 0  0 z 1 0  0 1  is an Identity matrix of order  and  0 21x 20and   1 0    0 01 y100z0 00 1      0  0 1    0 1 0 0 1 0  6   01 1 0  0  4 2 6   0 0 1  ve 3 4x 3,2respectively.  0  0 10 01 0  0 1  1 0 0 0 5 3   0 5 1 3 0  0101 1010 0004 2 60 1 0   4 2 6  40 1.1.3.10 2 6    4 2 below 6  0 0 50 0 361  Upper Triangular   matrix- A square 4020 2061610 5 3  0 matrix0 6 in which each 0element  5 3   4   principal the  diagonal  0is0 0,01is called 10 an upper 0 triangular 0 1  matrix. 0 5 3  0 05 53  30 0 6  0  0 6    0 4 0 2 6 6  041 020 160   4 0 0   0 0 6  ni  4 0 0    0   0 4 0 20 6  6 6    4 2  6     0Example 5 3  1.1.10   0 0 51 304is 0called 0an upper  2 5triangular 0  matrix oforder 4 0 3 x03.  42 05 0   04 52 36  4 0 4000 0000 0612 5 0 0 5   3 3 4 6    2  5 0  0 23 0 54 0 066  4   00 05 63  2 5  0  U     0 0 6   4 6 above   2 triangular 25 50  03matrix- 4 6A square which each 3element  3 41.1.3.11 4 0 6 0 Lower  0   4 4 2 06  0 6   a matrix c e in   3 4 6  the a principal c e  diagonal     3 is 4 0,0 4 is called  0 6   a lower4triangular 0  0 matrix.   2 5 0   3 240 565 03a c e b d f  ab cd ef   24 50 00  2 5 0  a c e  b ad cf  e  ity b 3 d 4 f 6  a3 0 c40 e66b d f     Example  a c  1.1.11 32 45 60 is a lower e    4a 6 b matrix of order 3 x 3.b d f   3 triangular a b  b bd d f  f  c d   ca d c e   3a 4 4c0 6e0a b    a b  a b  a c e  a c e e f   c aas b ceb1.1.4 d Equal  f  Matrices-a abb2 bd5Two f0cmatrices d  A and B are  said to be, written  d  A = B, if df  ba dand c fe corresponding  b d elements   are  of the same f are equal.  e  cf  d  m they  c type  c d3d 4 their  6e f      e a f b    ba  db f   a c e   e f   a c e   e ae f bf  a bb d f  ab c cd d ef  c a d c e a  c e   a c e   Example  1.1.12ca db   and c d  are not equal but  a c  eand )A aeb cfd e bf  d f     b e d f f   b  d f    a   ec fd    c e    e fa  c e   b d f  a c e  b bd d f  f    aba cd c ef eare  equal  eamatrix. a cf b ea c e  b d f  a c e      a c e a c e  b ad cf  e  bb d d f f  a abcc cdedefb d f      ba dc fe b d f   b d f  (c  b bde d fff     a c e   ba dc  ef  b d f  a a c c e e a c e    bab ddc fef  b d f   b d f    Amity Directorate of Distance & Online Education b d f     a c e  b d f     6 Business Mathematics Summary: Notes e An arrangement of m x n numbers or functions in the form of m horizontal lines (called rows) and n vertical lines (called columns), is called a matrix of in the type m by n (or m x n). A matrix having only one row is called a row matrix while the one having only one column is known as a column matrix. nl A m x n matrix each of whose elements is 0, is called a null matrix of the type m x n. A m x n matrix for which m = n, i.e., the number of rows equals the number of O columns is called a square matrix of order n or an n- rowed square matrix. A square matrix in which each one of the non-diagonal matrices is 0, is called a diagonal Matrix. ity A m x n matrix in which the number of rows and columns is uneven is called a rectangular matrix. A diagonal matrix in which all the diagonal elements are equal is called a scaler matrix. rs A square matrix is called a symmetric matrix if for every value of i and j aij = aji. Scalar matrix, each of whose diagonal element is unity, is called a unit matrix or an identity matrix. It is represented by ‘I’. ve A square matrix in which each element below the principal diagonal is 0, is called an upper triangular matrix. A square matrix in which each element above the principal diagonal is 0, is called a lower triangular matrix. ni Two matrices A and B are said to be, written as A = B, if they are of the same type and their corresponding elements are equal. U ity m )A (c Amity Directorate of Distance & Online Education  Business Mathematics 7 Unit-1.2: Algebra of Matrices Notes e Recall Session: in In the previous unit, you studied about: (a) Define matrix nl (b) Describe various types of matrices (c) Define equal matrices O Unit Outcome: At the end of this unit, you will be able to: Describe algebra of matrices ity Describe properties of matrix algebra Define transpose of a matrix Describe the properties of the transpose of a matrix 1.2.1 Introduction rs In the previous unit, we studied the various types of matrices and equal matrices ve as well. In this unit, we will learn about matrix algebra, properties of matrix algebra, the transpose of a matrix, and the properties of the transpose of a matrix. 1.2.2 Algebra of Matrices ni 1.2.2.1 Addition of Matrices If two matrices A and B are equal in number of rows and the number of columns U is also equal, then A + B is the matrix whose each element is equal to the sum of the corresponding elements of matrices A and B. Thus if a a  c c  A   1 a21 , aB2  1  c21 c2  ity  bA1  bb2  b  , dB1  dd2  d   1 2  1 2 AABa1 a1a2 ca1 ac2c1ca2 c2 c  then, A bB  d, B  1 b  d   1 2 2  b1 1b2 b1 1  d21 d1 b22d2d2  m 1.2.2.2  a  c a  c2  Difference A B  1 of1 Two2 Matrices b  d If twoamatrices b d   a1  A1 and2cB are 2c  equal  in number of rows and the number of columns is )A A   1 a21 , aB2  1  c21 c2  also equal, bA1then dB matrix  b2A– B is ,the b1 b2  1of Ad2and d  whose each element is equal to the difference of the dB.2 Thus if corresponding components 1 AABa1 a1a2 ca1 ac2c1ca2 c2 c  Ab bB1b  d, B b  d   1 1 2 2  1 2 b1 1  d21d1 b2 2d2d2  (c  a1  c1 a2  c2  AB  a ba  d b c d c  A   1  1a21 , B1a22 1 2 c21 c2  Amity Directorate of Distance & Online Education  bA1  bb2  b ,dB1  dd2  d   1 2  1 2 AB aa11c1a2aac2d1 aadc11c2 aca2 c2d2a d  A ABb c  b ,dB  b c  bd  1 1 2 1 1 2 2 2 b11 1 b2  2 1 d11 2 d2 2 2     2  1 2 b1  dA1  b2  d2  , B   1   a a    c ac   aca112 b b c a c 2  c2  d1 d2  A  B   a1  c1 a2  c2   a 1 c 11 a  c 2 2 b  d b  d  AAABBBA  B1  1 1 2 2  bbb11ddbd11bdbb212dadbd22cd2  a  c  1 2 2   1 1 2 2  1 A1  B 2 21  1 2 2 a a   c c   A   1 2  , B   1 b1 2d1 b2  d2  8  b1 b2  d1 d2  Business Mathematics  a a2  c1 c2   aaa11 aaa122 a2  ccc11 ccc122 c2  A   1  , B  d d  Notes AAAA bba1bbc,1,Bb, BBa2, Bddc2 dd d  1 2 1 2  b1 b2   1 2 e then, A  B b 1 11 b2 122 2 a1d1 11a2d2 122 2 c1 c2  b1  dA1  b2  d2   , B   a  c a   b a c 1 c b2  c d1 d2  a  a1  c1 a2  c2   a   c a  c222 2  A B in  a 1 c 111 a 12 c   b  d b  d  AAA BBB A B1 of  1 2 1.2.2.3 Multiplication  Two Matrices dbd22cd2  a  c  1 2 2 bbb11  dbd11 bdbb212 a  1 1 2 2  d d 21  1 2 2 If A and B are a1 a2  two 1 suchA 1  B 2   c1 c2  matrices that the number  of columns of A is equal to the A  number of rowsof B, then , B   the multiplication b  d  1 d  1 of2 A and d b  d 2  B matrix will be AB. nl  b1 b2   1 2 aaa11 aaa122 a2 ccc11 ccc122 c2  a1 a2   c1 c2   A    , B  d d  Thus ifAAA A bab 1 1c1 b a,2B,d,B1B,aB1c2  a2d2   2 1 2 b1 b2  b b AB b1 11 b2 122 2 ad11 11ad2 2122 2 c1 c2  d d d d d   1 2 O  b1c1  bA2d1  b1c2  b2,dB2     a a cc  a   aca d  d ab21daa1ccb2 2 a1aca2dd2 a2dd12  d2  AB  a1c1  a2d1 a1c2  a2d2  AB a1c11 1 a2d21 1 a1c12 2 a2d22 2 1 1 1 2 1 1 b c  b d b c  b d  then, AB AAB a AB bijbbc11cc1 b bc d b db c b bc db d   1 1 2 1 1 2 2 2  bd212d11 b 2ab c1c1c22ba 1b d2d2d222a 2c  a d   1 1 1 b1AB 2 1   11 21 22 21 1 2 2 2  ity  A aa .  b1c1  b2d1 b1c2  b2d2 A   aij  AAAaA 1.2.2.4 Negative aijijaij  ij ij Matrix A negative matrix a1 Aa A   a  adenoted 2 .a . ij  of a2 by – A where  A aij . AAAA  b aaijaijij. .ijA  1    1 b2   A ab1 . b2  a1 a2   a1 a2  rs  aaa11 aaa122 a2  aijaa11 aa1aa22a2  A     A   b b  Thus if AA A  aA bbc bbe,bthen 1 2 AAA  A   1 b b  bbb   2 b  b1 b2   1 2 A  b1 11 b2 122 2 a1ba1211 b12 22 2a1 a2  b d fA    A   b b  a c e a c a e c  eb b   1 2       ve 1.2.3 Properties  a  a cof c Matrix e e   1 Addition 2 A  A  A  A A abb bddb fdf f  Suppose A,bB,   a  c e b d f   d f  C three matrices are  of same order. A '   c d . A   (i) Commutative  a b aLaw:  b A b+ B d= B f+A a b  aea bf b  c'  Law: dc. A +.(B + C) = (A + B) + C A '   c d . ni AA ' (ii) Associative '   A A '  ' c d.  a b  c d. d  A' Identity: A    e f  (iii) Additive  eee ffefAA'f+ Oc = Ad =.O + A  kA   e of forder  m + n.   ' ' where O ' is kA a' ' null matrix A ' A U '  AA'A'' AA'AA  A ' (iv) Additive Inverse: A +' (– C) = O  ' ' 'A ' B A''  A   ' ' A  B kA  kA ' – AkA kA iskA kA called ' kA kA 'the kA '' kA  ' additive inverse of A. '  kA IfA+kA ' ity  AB   A  B A' B' ' ' (v) Cancellation  B ' ''A Law: ' B= ' A+C A A    A  B A' B' B A B '     B A A '' BAB ' ' B ' than B = C  A  B  A ' B ' '  AB   B ' A' '     '' '  AB   Bof' AScalar AB (vi) PropertiesAB ' AB   B B ' 'A A ' ' B' ' A 'Multiplication: k(A + B) = kA + kB and (k1 + k2) A = k1 A + k2 A, where k, k1 and  AB2arek  Bconstant or scaler quantity. ' ' A' m 1.2.4 Properties of Multiplication of Matrices Suppose A, B, C are three matrices. )A (i) Associative Law: (AB).C = A.(BC) (ii) Distributive Law: (a) A(B + C) = AB + BC (c (b) (A + B) C = AC + BC (iii) Identity Law: | A = A | = A Amity Directorate of Distance & Online Education  A   aij  a1c1  a2d1 a1c2 a2d 2 AB   b2d2 a .  b1c1  b2d1 b1c2A  ij  A   aij   a a   a1 a2  Business Mathematics A   1 2  A  9  b b2   A of aija.Matrix  b1 b2   1 1.2.5 Transpose Notes e The of a matrix is a matrix formed,a c e a1 a2  Aa1  from a2 the original  where the rows of the original A column matrix are the  of the A  transpose b d The  b matrix. f  transpose of matrix A is denoted by in A′.  b1 b2   1 b2  a b  a c e  Thus if A    then A '   c d . nl b d f     e f  a b   A'of  Matrices ' A A '   c ofdTranspose 1.2.6 Properties . O  (a) (A′) = A  e f   kA  kA' ' (b) (kA) = kA′  A' ' A  A  B ' A ' B ' ity (c) (A + B) = A′ + B′  kA=B′A′ ' (d) (AB)′  kA '  AB '  B' A' Summary: A  B  A ' B ' ' If two matrices  AB   B ' AA ' rs ' and B are equal in the number of rows and the number of columns is also equal, then A + B is the matrix whose each element is equal to the sum of the corresponding elements of matrices A and B. ve If two matrices A and B are equal in the number of rows and the number of columns is also equal, then A – B is the matrix whose each element is equal to the difference of the corresponding components of A and B. If A and B are two such matrices that the number of columns of A is equal to ni the number of rows of B, then the multiplication of A and B matrix will be AB. The negative form of the matrix A = – A. U The of a matrix is a matrix formed, from the original where the rows of the original matrix are the column of the transpose matrix. The transpose of matrix A is denoted by A′. ity m )A (c Amity Directorate of Distance & Online Education  10 Business Mathematics Unit - 1.3: Determinants Notes e Recall Session: in In the previous unit, you studied about: (a) Describe algebra of matrices nl (b) Describe properties of matrix algebra (c) Define transpose of a matrix (d) Describe the properties of the transpose of a matrix O Unit Outcome: At the end of this unit, you will be able to: ity Define determinants Describe prope