3.4 If \( \mathbf{A}=\left[\begin{array}{cc}2 & -5 \\ 3 & -1 \\ -1 & 1\end{array}\right] \) and \( \mathbf{B}=\left[\begin{array}{lll}1 & 0 & 5\end{array}\right] \) (a) Determine the size of \( \mathbf{A} \) and \( \mathbf{B} \) (b) Evaluate \( \mathbf{B A} \)

To determine the size of matrix \( \mathbf{A} \) and matrix \( \mathbf{B} \), we need to find the number of rows and columns in each matrix. Given: \[ \mathbf{A}=\left[\begin{array}{cc}2 & -5 \\ 3 & -1 \\ -1 & 1\end{array}\right] \] \[ \mathbf{B}=\left[\begin{array}{lll}1 & 0 & 5\end{array}\right] \] (a) The size of matrix \( \mathbf{A} \) is \( 3 \times 2 \) because it has 3 rows and 2 columns. The size of matrix \( \mathbf{B} \) is \( 1 \times 3 \) because it has 1 row and 3 columns. (b) To evaluate \( \mathbf{B A} \), we need to perform matrix multiplication. The result will be a matrix with the same number of rows as matrix \( \mathbf{B} \) and the same number of columns as matrix \( \mathbf{A} \). Let's calculate \( \mathbf{B A} \) step by step. Matrices by following steps: - step0: Solution: \(\left[\begin{array}{rrr}{1}&{0}&{5}\end{array}\right]\times \left[\begin{array}{rr}{2}&{-5}\\{3}&{-1}\\{-1}&{1}\end{array}\right]\) - step1: Multiply the terms: \(\left[\begin{array}{rr}{1\times 2+0\times 3+5\left(-1\right)}&{1\times \left(-5\right)+0\times \left(-1\right)+5\times 1}\end{array}\right]\) - step2: Evaluate: \(\left[\begin{array}{rr}{-3}&{0}\end{array}\right]\) The result of \( \mathbf{B A} \) is \( \left[\begin{array}{rr}{-3} & {0}\end{array}\right] \).