Questions: [ left[beginarraycc 5 -7 4 6 endarrayright]left[beginarrayc 4 -8 endarrayright] ]

Solution Steps

To find the product of a matrix and a vector, we perform matrix multiplication. This involves taking the dot product of each row of the matrix with the vector. For a matrix \( A \) with dimensions \( m \times n \) and a vector \( v \) with dimensions \( n \times 1 \), the resulting product will be a vector with dimensions \( m \times 1 \).

We are given a matrix \( A \) and a vector \( v \). The matrix \( A \) is: \[ A = \begin{bmatrix} 5 & -7 \\ 4 & 6 \end{bmatrix} \] and the vector \( v \) is: \[ v = \begin{bmatrix} 4 \\ -8 \end{bmatrix} \]

To find the product \( Av \), we perform the matrix-vector multiplication. This involves taking the dot product of each row of the matrix with the vector.

• For the first row of \( A \): \[ 5 \times 4 + (-7) \times (-8) = 20 + 56 = 76 \]

• For the second row of \( A \): \[ 4 \times 4 + 6 \times (-8) = 16 - 48 = -32 \]

Final Answer