Problem 64 - College Algebra

Question

Find two matrices \(A\) and \(B\) such that \(A B=B A\)

Step-by-Step Solution

Answer

So, two matrices \(A\) and \(B\) that satisfy \(A B=B A\) are \(A = \[ \[1, 0\], \[0, 1\] \] \) and \(B = \[ \[2, 0\], \[0, 2\] \]\).

Step 1: Choose Matrix A

Let's choose a simple diagonal matrix for \(A\). For instance, \(A = \[ \[1, 0\], \[0, 1\] \] \). This is the 2x2 Identity matrix.

Step 2: Choose Matrix B

Likewise, for \(B\), let's also choose a diagonal matrix. For instance, \(B = \[ \[2, 0\], \[0, 2\] \] \). This is a scalar matrix, meaning all its entries are the same.

Step 3: Verify the Commutative Property

Now, let's multiply \(A\) and \(B\) and \(B\) and \(A\) to see if they are indeed equal. \(A \times B = \[ \[1, 0\], \[0, 1\] \] \times \[ \[2, 0\], \[0, 2\] \] = \[ \[2, 0\], \[0, 2\] \] \) and \(B \times A = \[ \[2, 0\], \[0, 2\] \] \times \[ \[1, 0\], \[0, 1\] \] = \[ \[2, 0\], \[0, 2\] \] \). So, both are equal, which satisfies the problem statement.