Let \( A=\left[\begin{array}{cc}-5 & 0 \\ 0 & 4\end{array}\right] \) and \( B=\left[\begin{array}{cc}-2 & 0 \\ 0 & 1\end{array}\right] \). Find each matrix below. If a matrix is not defined, click on "Undefined". (a) \( B A= \) \( \square \) (b) \( A B= \) \( \square \) (c) \( B^{2}- \) \( \square \) Check

Let's compute each of the given matrix operations step by step. ### Given Matrices: \[ A = \begin{bmatrix} -5 & 0 \\ 0 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} -2 & 0 \\ 0 & 1 \end{bmatrix} \] ### (a) \( BA \) Since both \( A \) and \( B \) are diagonal matrices, their product is also a diagonal matrix where each diagonal element is the product of the corresponding diagonal elements of \( B \) and \( A \). \[ BA = B \cdot A = \begin{bmatrix} (-2)(-5) & 0 \cdot 0 \\ 0 \cdot 0 & 1 \cdot 4 \end{bmatrix} = \begin{bmatrix} 10 & 0 \\ 0 & 4 \end{bmatrix} \] ### (b) \( AB \) Similarly, multiplying \( A \) by \( B \): \[ AB = A \cdot B = \begin{bmatrix} (-5)(-2) & 0 \cdot 0 \\ 0 \cdot 0 & 4 \cdot 1 \end{bmatrix} = \begin{bmatrix} 10 & 0 \\ 0 & 4 \end{bmatrix} \] ### (c) \( B^2 \) To find \( B^2 \), multiply \( B \) by itself: \[ B^2 = B \cdot B = \begin{bmatrix} (-2)(-2) & 0 \cdot 0 \\ 0 \cdot 0 & 1 \cdot 1 \end{bmatrix} = \begin{bmatrix} 4 & 0 \\ 0 & 1 \end{bmatrix} \] ### Summary of Results: - **(a) \( BA = \begin{bmatrix} 10 & 0 \\ 0 & 4 \end{bmatrix} \)** - **(b) \( AB = \begin{bmatrix} 10 & 0 \\ 0 & 4 \end{bmatrix} \)** - **(c) \( B^2 = \begin{bmatrix} 4 & 0 \\ 0 & 1 \end{bmatrix} \)**