1. **Matrix multiplication problem:** Given matrices $$ A = \begin{bmatrix} 1 & 0 \\ 1 & 0 \\ 1 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix} $$ We want to find the product $AB$. 2. **Matrix multiplication rule:** The product $AB$ is defined if the number of columns of $A$ equals the number of rows of $B$. Here, $A$ is $3 \times 2$ and $B$ is $2 \times 2$, so multiplication is possible and the result will be a $3 \times 2$ matrix. 3. **Calculate each element of $AB$:** The element in row $i$, column $j$ of $AB$ is computed as the dot product of row $i$ of $A$ and column $j$ of $B$. 4. Compute elements: - $(AB)_{11} = 1 \times 0 + 0 \times 0 = 0$ - $(AB)_{12} = 1 \times 1 + 0 \times 1 = 1$ - $(AB)_{21} = 1 \times 0 + 0 \times 0 = 0$ - $(AB)_{22} = 1 \times 1 + 0 \times 1 = 1$ - $(AB)_{31} = 1 \times 0 + 0 \times 0 = 0$ - $(AB)_{32} = 1 \times 1 + 0 \times 1 = 1$ 5. So, $$ AB = \begin{bmatrix} 0 & 1 \\ 0 & 1 \\ 0 & 1 \end{bmatrix} $$ This matches option B. --- 6. **If $4Z$ is a scalar matrix, what is $Z$?** 7. A scalar matrix is a diagonal matrix with all diagonal entries equal. If $4Z$ is scalar, then $Z$ must be a diagonal matrix (since scalar matrices are a subset of diagonal matrices). So the answer is C. --- 8. **Matrix $A$ is:** $$ A = \begin{bmatrix} 2 & 0 & -4 \\ 4 & -2 & -2 \\ 1 & 2 & 6 \end{bmatrix} $$ 9. **Minor of element $a_{23}$:** - $a_{23}$ is element in 2nd row, 3rd column, which is $-2$. - Minor is determinant of matrix formed by deleting 2nd row and 3rd column: $$ \begin{bmatrix} 2 & 0 \\ 1 & 2 \end{bmatrix} $$ - Determinant: $$ 2 \times 2 - 0 \times 1 = 4 $$ So minor is 4, option D. --- 10. **Cofactor of element $A_{32}$:** - $A_{32}$ is element in 3rd row, 2nd column, which is 2. - Minor is determinant of matrix formed by deleting 3rd row and 2nd column: $$ \begin{bmatrix} 2 & -4 \\ 4 & -2 \end{bmatrix} $$ - Determinant: $$ 2 \times (-2) - (-4) \times 4 = -4 + 16 = 12 $$ - Cofactor is $(-1)^{3+2} \times 12 = (-1)^5 \times 12 = -12$ Final answers: - Q6: B - Q7: C - Q8: D - Q9: Cofactor of $A_{32}$ is $-12$.