Matrix Multiplication-show all workl \( A=\left[\begin{array}{ll}7 & 3 \\ 6 & 1\end{array}\right] \quad B=\left[\begin{array}{ll}4 & 2 \\ 2 & 6\end{array}\right] \) \( A B= \) \( B A= \)

We are given: \[ A=\begin{bmatrix} 7 & 3 \\ 6 & 1 \end{bmatrix}, \quad B=\begin{bmatrix} 4 & 2 \\ 2 & 6 \end{bmatrix} \] **Step 1: Compute \( AB \)** To find \( AB \), we use the formula for matrix multiplication: \[ AB = \begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \\ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{bmatrix} \] For our matrices: \[ AB = \begin{bmatrix} 7\cdot 4 + 3\cdot 2 & 7\cdot 2 + 3\cdot 6 \\ 6\cdot 4 + 1\cdot 2 & 6\cdot 2 + 1\cdot 6 \end{bmatrix} \] Now, compute each entry step by step: - Top-left: \[ 7\cdot4 + 3\cdot2 = 28 + 6 = 34 \] - Top-right: \[ 7\cdot2 + 3\cdot6 = 14 + 18 = 32 \] - Bottom-left: \[ 6\cdot4 + 1\cdot2 = 24 + 2 = 26 \] - Bottom-right: \[ 6\cdot2 + 1\cdot6 = 12 + 6 = 18 \] Thus, the product \( AB \) is: \[ AB = \begin{bmatrix} 34 & 32 \\ 26 & 18 \end{bmatrix} \] **Step 2: Compute \( BA \)** Next, to compute \( BA \), we use the multiplication rule again: \[ BA = \begin{bmatrix} b_{11}a_{11} + b_{12}a_{21} & b_{11}a_{12} + b_{12}a_{22} \\ b_{21}a_{11} + b_{22}a_{21} & b_{21}a_{12} + b_{22}a_{22} \end{bmatrix} \] For our matrices, this gives: \[ BA = \begin{bmatrix} 4\cdot7 + 2\cdot6 & 4\cdot3 + 2\cdot1 \\ 2\cdot7 + 6\cdot6 & 2\cdot3 + 6\cdot1 \end{bmatrix} \] Now, compute each entry: - Top-left: \[ 4\cdot7 + 2\cdot6 = 28 + 12 = 40 \] - Top-right: \[ 4\cdot3 + 2\cdot1 = 12 + 2 = 14 \] - Bottom-left: \[ 2\cdot7 + 6\cdot6 = 14 + 36 = 50 \] - Bottom-right: \[ 2\cdot3 + 6\cdot1 = 6 + 6 = 12 \] Thus, the product \( BA \) is: \[ BA = \begin{bmatrix} 40 & 14 \\ 50 & 12 \end{bmatrix} \]