MM1_Chapter_10

Lemma 1019 multiplication by the identity matrix let

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Unformatted text preview: ×1+5×0 3×0+1×1 2×0+5×1 = 31 25 . and I2 A = We observe that I2 A = A I2 = A. 2 The identity I2 A = A I2 = A which we observed in the last example is true for every 2 × 2 matrix A. Moreover, an analogous property holds for the n × n identity matrix. Lemma 10.19 (multiplication by the identity matrix) Let A be an m × n matrix, let B be a n × ℓ matrix, and let In be the n × n identity matrix. Then we have A In = A and In B = B. (10.1) In particular, if A is an n × n square matrix, then (10.1) implies A In = In A = A. We observe that In plays the role in the multiplication of n × n matrices that the real number 1 plays in the multiplication of real numbers, that is, x1 = 1x = x for any real number x ∈ R. 272 10. Matrices Theorem 10.20 (associative law of matrix multiplication) Let A be an m × n matrix, B an n × ℓ matrix, and C an ℓ × k matrix. Then (A B ) C = A (B C ). Example 10.21 (associative law of matrix multiplication) Verify the associative law for the three matrices 31 25 A= , 1 −1 −1 1 B= , 12 21 C= . Solution: We have have AB = 31 25 1 −1 −1 1 = 3 − 1 −3 + 1 2 − 5 −2 + 5 12 21 = 2−4 4−2 −3 + 6 −6 + 3 2 −2 −3 3 = and (A B ) C = 2 −2 −3 3 = −2 2 3 −3 . On the other hand BC = 1 −1 −1 1 12 21 A (B C ) = 31 25 −1 1 1 −1 1−2 2−1 −1 + 2 −2 + 1 = = −1 1 1 −1 −3 + 1 3 − 1 −2 + 5 2 − 5 = −2 2 3 −3 and = and we find indeed (A B ) C = A (B C ). Definition 10.22 (k -fold product of a matrix) For any integer k ∈ N, we denote by Ak the k -fold product of A, that is, Ak = A A . . . A k -times Finally we introduce the transpose of a matrix. , 2 10. Matrices 273 Definition 10.23 (transpose of a matrix) The transpose AT of an m × n matrix A is the n × m matrix AT obtained by interchanging the rows and columns of A. In formulas, [AT ]i,j = aj,i for all 1 ≤ i ≤ n and 1 ≤ j ≤ m. (10.2) Example 10.24 (transpose of a matrix) A= 231 210 22 AT = 3 1 . 10 and Remark 10.25 (transpose of transpose gives original matrix) For any matrix A, we have (AT )T = A. Proof: Indeed, if A is an m × n matrix, then AT is an n × m matrix, and so (AT )T is an m × n matrix. Furthermore, we have from (10.2) that (AT )T 10.3 i,j = AT j,i = ai,j for all 1 ≤ i ≤ m and all 1 ≤ j ≤ n. 2 The Determinant of a Square Matrix The determinant associates a scalar with a matrix and can only be defined for square matrices. We will start by discussing the determinant for the special cases of 2 × 2 matrices and 3 × 3 matrices. After that we will define the determinant recursively for n × n matrices. We will learn several properties of determinants that can be used to simplify the computation of determinants. Definition 10.26 (determinant of 2 × 2 matrix) The determinant det(A) of a 2 × 2 matrix A= ab cd is defined by det(A) = ab cd = a d − b c. (10.3) 274 10. Matrices Example 10.27 (determinants of 2 × 2 matrices) Compute the determinants of the 2 × 2 matrices A= 13 24 and 25 4 10 B= . Solution: From (10.3), we have that det(A) = det(B ) = 13 24 25 4 10 = 1 × 4 − 3 × 2 = 4 − 6 = −2, = 2 × 10 − 5 × 4 = 20 − 20 = 0. 2 Definition 10.28 (determinant of a 3 × 3 matrix) The det...
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