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Chapter 8: Problem 48

If possible, find (a) \(A B,(b) B A,\) and \((c) A^{2}\). $$A=\left[\begin{array}{ll} 1 & 2 \\ 4 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -1 \\ -1 & 8 \end{array}\right]$$

Short Answer

Expert verified
\(AB = [0, 15; 6, 12]\), \(BA = [-2, 0; 30, 12]\), and \(A^2 = [9, 6; 12, 8]\)

Step by step solution

01

Multiplication of A and B

To find \(AB\), we multiply matrix A by matrix B. This involves taking the dot product of the rows of A with the columns of B as follows: \(AB = [(1*2 + 2*-1), (1*-1 + 2*8); (4*2 + 2*-1), (4*-1 + 2*8)] = [0, 15; 6, 12]\)
02

Multiplication of B and A

In the case of \(BA\), we multiply matrix B by matrix A. This is calculated as: \(BA = [(2*1 + -1*4), (2*2 + -1*2); (-1*1 + 8*4), (-1*2 + 8*2)] = [-2, 0; 30, 12]\)
03

Square of A

To find \(A^2\), we multiply the matrix A by itself. We compute this as: \(A^2= [(1*1 + 2*4), (1*2 + 2*2); (4*1 + 2*4), (4*2 + 2*2)] = [9, 6; 12, 8]\)

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