Questions: A=[5 0 0; 0 1 0; 0 0 3] and B=[6 0 0; 0 4 0; 0 0 8]. AB=[42 0 0; 0 □ 0; 0 0 □]

Solution Steps

To find the product of two diagonal matrices \( A \) and \( B \), we multiply the corresponding diagonal elements. Since all off-diagonal elements are zero, the resulting matrix will also be a diagonal matrix. Therefore, the element at position (i, i) in the product matrix \( AB \) is simply the product of the elements at position (i, i) in matrices \( A \) and \( B \).

We have two diagonal matrices defined as follows: \[ A = \begin{bmatrix} 5 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 6 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 8 \end{bmatrix} \]

To find the product \( AB \), we multiply the corresponding diagonal elements: \[ AB = \begin{bmatrix} 5 \cdot 6 & 0 & 0 \\ 0 & 1 \cdot 4 & 0 \\ 0 & 0 & 3 \cdot 8 \end{bmatrix} = \begin{bmatrix} 30 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 24 \end{bmatrix} \]

The resulting matrix from the multiplication is: \[ AB = \begin{bmatrix} 30 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 24 \end{bmatrix} \]

Final Answer