Linear Systems and Matrices PDF

# **Linear Systems and Matrices** ## **5.1 Definition of a Matrix** - A matrix is a set of *mn* quantities arranged in a rectangular array of *m* rows and *n* columns. - When writing down matrices it is useful to enclose this array by large brackets and denote the matrix by a single letter, say *A...

# **Linear Systems and Matrices** ## **5.1 Definition of a Matrix** - A matrix is a set of *mn* quantities arranged in a rectangular array of *m* rows and *n* columns. - When writing down matrices it is useful to enclose this array by large brackets and denote the matrix by a single letter, say *A*. - This is the general form of a matrix: A = ``` a11 a12 ........ a1n a21 a22 ........ a2n .................... am1 am2 ........ amn ``` - The elements of a matrix, *a_ik*, are called the elements of the matrix. - A matrix of *m* rows and *n* columns is said to be of order (*m x n*), and to be square if *m* = *n*. ## **5.2 Algebra of Matrices** ### **(a) Addition and Subtraction** - To execute addition and subtraction, matrices must be of the same order. - If *A* and *B* are two matrices of the same order with elements *a_ik* and *b_ik*, their sum *A + B* is defined as a matrix *C* whose elements *c_ik = a_ik + b_ik*. - *C* has the same order as *A* and *B*. For example, if: A = ``` (1 2 3) (0 -1 2) ``` and, B = ``` (3 4 5) (1 2 3) ``` Then: A + B = ``` (1+3 2+4 3+5) (0+1 -1+2 2+3) ``` = ``` (4 6 8) (1 1 5) ``` C = A + B = ``` ( 4 6 8) ( 1 1 5) ``` Subtraction of matrices is defined in a similar manner : A - B = ``` (1-3 2-4 3-5) (0-1 -1-2 2-3) ``` = ``` (-2 -2 -2) (-1 -3 -1 ) ``` Also: - A + B = B + A: Associative - (A + B) + C = A + (B + C): Commutative ### **(b) Equality of Matrices** - Two matrices *A* and *B* with elements *a_ik* and *b_ik* respectively are equal only if they are of the same order, and if all their corresponding elements are equal (*i.e.* a_ik = b_ik). ### **(c) Multiplication by a Number** Example: 3A = ``` (3 * 1 3 * 2 3 * 3) (3 * 2 3 * 3 3 * 4) ``` = ``` ( 3 6 9) ( 6 9 12) ``` ## **5.3 Special Types of Matrices** ### **(a) Null-Matrix** - Any matrix of order (*m x n*) with all its elements equal to zero is called a null matrix of order (*m x n*). N = ``` (0 0) (0 0) ``` ### **(b) Unit matrix** - A square matrix with zero elements everywhere except in the diagonal is a unit matrix. - The general form is: I = ``` (1 0) (0 1) ``` In general: IA = AI = A I = I² = I³ = ...... = I^k ### **(d) Matrix Multiplication** - The definition of matrix multiplication is such that two matrices *A* and *B* can only be multiplied together to form their product *AB* when the number of columns of *A* is equal to the number of rows of *B*. - Suppose *A* is a matrix of order (*m x p*) with elements *a_ik*, and *B* is a matrix of order (*p x n*) with elements *b_ik*. - Then their product *AB* is a matrix *C* of order (*m x n*) with elements *c_ik* defined by: ``` C<sub>ik</sub> = ∑<sub>s=1</sub><sup>p</sup> a<sub>is</sub> b<sub>sk</sub> ``` For example, if *A* and *B* are (2 x 2) and (2 x 2) matrices respectively given by: A = ``` (a<sub>11</sub> a<sub>12</sub>) (a<sub>21</sub> a<sub>22</sub>) ``` B = ``` (b<sub>11</sub> b<sub>12</sub>) (b<sub>21</sub> b<sub>22</sub>) ``` Then the product C = AB is a (2 x 2) matrix defined as: C = AB = ``` (a<sub>11</sub>b<sub>11</sub> + a<sub>12</sub>b<sub>21</sub> a<sub>11</sub>b<sub>12</sub> + a<sub>12</sub>b<sub>22</sub>) (a<sub>21</sub>b<sub>11</sub> + a<sub>22</sub>b<sub>21</sub> a<sub>21</sub>b<sub>12</sub> + a<sub>22</sub>b<sub>22</sub>) ``` ### **Examples** - A = ``` (1 2) (3 4) ``` B = ``` (2 3) (1 2) ``` Find AB and BA. - AB = ``` (1*2 + 2 * 1 1 * 3 + 2 * 2) (3 * 2 + 4 * 1 3 * 3 + 4 * 2) ``` = ``` (4 7) (10 17) ``` - BA = ``` (2 * 1 + 3 * 3 2 * 2 + 3 * 4) (1 * 1 + 2 * 3 1 * 2 + 2 * 4) ``` = ``` (11 16) (7 10) ``` - A = ``` (2 5) (4 10) ``` B = ``` (3 -1) (1 3) ``` Find AB. - AB = ``` (2 * 3 + 5 * 1 2 * -1 + 5 * 3) (4 * 3 + 10 * 1 4 * -1 + 10 * 3) ``` = ``` (11 13) (22 26) ``` - A = ``` (1 2) (3 4) ``` B = ``` (2 0) (1 1) ``` Find *AB*. - AB = ``` (1*2 + 2 * 1 1 * 0 + 2 * 1) (3 * 2 + 4 * 1 3 * 0 + 4 * 1) ``` = ``` (4 2) (10 4) ``` - A = ``` (-1 2) (3 1) ``` B = ``` (3 -1) (1 -1) ``` Find *AB*. - AB = ``` ( -1 * 3 + 2 * 1 -1 * -1 + 2 * -1) (3 * 3 + 1 * 1 3 * -1 + 1 * -1) ``` = ``` (-1 -1) (10 -4) ``` - Clearly AB ≠ BA even though products are defined. This example shows that matrix multiplication is not commutative. - However, matrix multiplication is commutative in the associative and distributive laws of multiplication: - (AB)C = A(BC): Associative - (A+B)C = AC + BC: Distributive - Provided that the products are defined. - A = ``` (1 2) (0 1) ``` Find A², A³, 3A² + 3I. - A² = A * A = ``` (1 2) * (1 2) (0 1) (0 1) ``` = ``` (1 * 1 + 2 * 0 1 * 2 + 2 * 1) (0 * 1 + 1 * 0 0 * 2 + 1 * 1) ``` = ``` (1 4) (0 1) ``` - A³ = A * A² = ``` (1 2) * (1 4) (0 1) (0 1) ``` = ``` (1 * 1 + 2 * 0 1 * 4 + 2 * 1) (0 * 1 + 1 * 0 0 * 4 + 1 * 1) ``` = ``` (1 6) (0 1) ``` - 3A² + 3I = 3 * ``` (1 4) + 3 * ( 0 1) ``` = 3 * ``` (1 4) (0 1) ``` + 3 * ``` (1 0) (0 1) ``` = ``` (3 12) (0 3) ``` + ``` (3 0) (0 3) ``` = ``` (1 - 3+3 0-0+0) (6-12+0 1-3+3) ``` = ``` (1 0) (-6 1) ``` <start_of_image> - A = ``` (-1 2) (2 1) ``` Find (A - 2I)². - A - 2I = ``` (-1 2) - 2 * (2 1) ``` = ``` (-1 2) - (2 0) (0 2) ``` = ``` (-1 2) (-2 1) ``` - (A - 2I)² = (A - 2I) * (A - 2I) = ``` ( -1 2) * ( - 1 2) (-2 1) (-2 1) ``` = ``` ( -1 * -1 + 2 * -2 - 1 * 2 + 2 * 1) ( -2 * -1 + 1 * -2 -2 * 2 + 1 * 1) ``` = ``` (-3 0) ( 0 -3) ``` - A = ``` (2 -1) (3 1) ``` B = ``` (1 2) (-1 2) ``` Find A², B², (A+B)², (A-B)², A² - B², A² + B², A³ - 2A² + 3I, B⁴ + 3B² - 5I. - A² = A * A = ``` (2 -1) * (2 -1) (3 1) (3 1) ``` = ``` (2 * 2 + -1 * 3 2 * -1 + -1 * 1) ( 3 * 2 + 1 * 3 3 * -1 + 1 * 1) ``` = ``` (1 -3) (9 -2) ``` - B² = B * B = ``` (1 2) * (1 2) (-1 2) (-1 2) ``` = ``` (1*1 + 2 * -1 1 * 2 + 2 * 2) (-1 * 1 + 2 * -1 -1 * 2 + 2 * 2) ``` = ``` (-1 6) (-3 2) ``` - A + B = ``` (2 -1) + (1 2) (3 1) (-1 2) ``` = ``` (2+1 -1+2) (3-1 1+2) ``` = ``` (3 1) (2 3) ``` - (A + B)² = (A + B) * (A + B) = ``` (3 1) * (3 1) (2 3) (2 3) ``` = ``` (3 * 3 + 1 * 2 3 * 1 + 1 * 3) (2 * 3 + 3 * 2 2 * 1 + 3 * 3) ``` = ``` (11 6) (12 11) ``` - A³ = A. A² = ``` (2 -1) * (1 -3) (3 1) (9 - 2) ``` = ``` (2*1 + -1 *9 2*-3 + -1 * -2 ) (3*1 +1 * 9 3 * -3 + 1 * -2) ``` = ``` (-7 4) (12 -11) ``` - 2A² = 2 * ``` (1 -3) (9 -2) ``` = ``` (2 -6) (18 -4) ``` - 3I = 3 * ``` (1 0) (0 1) ``` = ``` (3 0) (0 3) ``` - A³ - 2A² + 3I = ``` (-7 4) ( 12 -11) ``` - ``` ( 2 -6) (18 -4) ``` + ``` (3 0) (0 3) ``` = ``` (-7 - 2 + 3 4 - 6 + 0) (12 - 18 + 0 -11 + 4 + 3) ``` = ``` (-6 2) (-6 -4) ``` - A = ``` (2 -1) (3 1) ``` - A² = ``` (1 -3) (9 -2) ``` - A³ = A * A² = ``` ( 2 -1) * (1 -3) ( 3 1) (9 -2) ``` = ``` (2 * 1 + -1 * 9 2 * -3 + -1 * -2) (3 * 1 + 1 * 9 3 * -3 + 1 * -2) ``` = ``` (-7 4) (12 -11) ```