MM1_Chapter_10

And the sum a b is given by ab 12 34 c let a be

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Unformatted text preview: two 2 × 2 matrices A= 12 34 12 34 − and B= 11 11 is given by A−B = 11 11 = 1−1 2−1 3−1 4−1 = 01 23 , 11 11 = 1+1 2+1 3+1 4+1 = 23 45 . and the sum A + B is given by A+B = 12 34 + (c) Let A be an arbitrary m × n matrix and let O be the m × n zero matrix. Then A + O = O + A = A. Indeed [A+O]i,j = ai,j +0 = [O+A]i,j = 0+ai,j = ai,j for all 1 ≤ i ≤ m and all 1 ≤ j ≤ n. 10. Matrices 269 (d) If A is a matrix and k is a positive integer, then A + A + . . . + A = k A. 2 k -times Next we learn that we can multiply an m × n matrix A with and n × ℓ matrix B , and the result will be an m × ℓ matrix. The matrix multiplication A B is rather more complicated than scalar multiplication or matrix addition, since matrix multiplication is not performed by multiplying corresponding entries but by taking the scalar products of rows the matrix A and columns of the matrix B . Definition 10.13 (multiplication of an m × n by an n × ℓ matrix) Let A = (ai,j ) be an m × n matrix, and let B = (bj,k ) be an n × ℓ matrix. Then the product A B of the m × n matrix A with the n × ℓ matrix B is defined to be the m × ℓ matrix A = ([A B ]i,j ) with the entries n ai,k bk,j (i, j )th entry of A B = [A B ]i,j = k =1 = scalar product of ith row vector of A and j th column vector of B a1,j a2,j = ai,1 , ai,2 , . . . , ai,n · . . . . an,j Note that the product A B only exists if the number of columns in the first matrix A equals the number of rows in the second matrix B . Note that if A and B are both n × n matrices, then both A B and B A exist but in general A B and B A are not necessarily equal. Example 10.14 (matrix multiplication) The matrices A and B , given by A= 31 25 and B= −2 6 47 . They are both 2 × 2 matrices, and we can compute both A B and B A. We find AB = 31 25 −2 6 47 270 10. Matrices = 3 × (−2) + 1 × 4 2 × (−2) + 5 × 4 = −2 25 16 47 3×6+1×7 2×6+5×7 , and 31 25 −2 6 47 BA = = (−2) × 3 + 6 × 2 4×3+7×2 = 6 28 26 39 (−2) × 1 + 6 × 5 4×1+7×5 . We see that A B = B A. 2 Example 10.15 (matrix multiplication) Consider the two matrices A = (1 2) 1 B = 2 . 3 and Then A B does not exist, since A is a 1 × 2 matrix and B is a 3 × 1 matrix. The product B A does exist, since B is a 3 × 1 and A is a 1 × 2 matrix. We find 1 1×1 1×2 12 B A = 2 (1 2) = 2 × 1 2 × 2 = 2 4 . 2 3 3×1 3×2 36 Definition 10.16 (identity matrix) The n × n identity matrix In is the n × n matrix which has ones on the leading diagonal and zeros everywhere else, that is, [In ]i,i = 1 for 1 ≤ i ≤ n and [In ]i,j = 0 if i = j. Explicitly the identity matrix In is In = 100 010 001 ... ... ... 000 ··· ··· ··· .. . 0 0 0 . . . ··· 1 . 10. Matrices 271 Example 10.17 (identity matrix) The 2 × 2 identity matrix and the 3 × 3 identity matrix are given by 100 10 and I3 = 0 1 0 , I2 = 01 001 respectively. 2 Example 10.18 (multiplication with the identity matrix) Let the 2 × 2 matrix A be given by A= 31 25 . Compute the products I2 A and A I2 , where I2 is the 2 × 2 identity matrix. Solution: I2 A = 10 01 31 25 = 1×3+0×2 0×3+1×2 1×1+0×5 0×1+1×5 = 31 25 , 31 25 10 01 = 3×1+1×0 2...
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This document was uploaded on 01/31/2014.

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