Question

Consider the matrices \[ A = \begin{bmatrix} x & y & 0 \\ -3 & 1 & 2 \\ 1 & -2 & z \end{bmatrix}, B = \begin{bmatrix} 1 & -2 & -2 \\ 2 & 0 & 1 \\ 2 & 1 & 0 \end{bmatrix} \] If the cofactors of the elements \( z, 1 \) in 3rd row and \( x \) of \( A \) are 9, 4, 3 respectively, then \( AB = \)

• A. \( \begin{bmatrix} -7 & -4 & -8 \\ -1 & 8 & 7 \\ 3 & -3 & -4 \end{bmatrix} \)
• B. \( \begin{bmatrix} 7 & -6 & 8 \\ -5 & 4 & -5 \\ -5 & -3 & -4 \end{bmatrix} \)
• C. \( \begin{bmatrix} 7 & -6 & -4 \\ 3 & 8 & 7 \\ -5 & -3 & -4 \end{bmatrix} \)
• D. \( \begin{bmatrix} 7 & -6 & 8 \\ -1 & 8 & -5 \\ 3 & -3 & -4 \end{bmatrix} \)

Hint:

To find matrix multiplication \( AB \), ensure all entries of A are known or determined first using the cofactor information. Then use row-by-column multiplication to compute the product.

Answer 1

The Correct Option is C

We are given the cofactors of elements \( z \), \( 1 \) in 3rd row, and \( x \) of matrix \( A \) as 9, 4, 3 respectively.

Use these to determine the values:

- The cofactor of \( A_{3,3} = z \) is 9.
- The cofactor of \( A_{3,2} = 1 \) is 4.
- The cofactor of \( A_{1,1} = x \) is 3.

Using cofactor expansion and determinant properties, these values allow solving for \( x, y, z \).

After substitution and matrix multiplication of \( A \cdot B \), we get:

\[ AB = \begin{bmatrix} 7 & -6 & -4 \\ 3 & 8 & 7 \\ -5 & -3 & -4 \end{bmatrix} \]