Question

The determinant of the matrix \( M \) shown below is \(\underline{\hspace{2cm}}\).  \[ M = \begin{bmatrix} 1 & 2 & 0 & 0 \\ 3 & 4 & 0 & 0 \\ 0 & 0 & 4 & 3 \\ 0 & 0 & 0 & 1 \end{bmatrix} \]

Hint:

To find the determinant of a block matrix, use cofactor expansion along a row or column that simplifies the calculation.

Answer 1

Correct Answer: 4

To compute the determinant of the matrix \( M \), we can use cofactor expansion along the last row: \[ \text{det}(M) = 1 \times \text{det}\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \times \text{det}\begin{pmatrix} 4 & 3 \\ 0 & 0 \end{pmatrix} \] First, calculate the determinant of the 2x2 matrix: \[ \text{det}\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = (1)(4) - (2)(3) = 4 - 6 = -2 \] Now, calculate the determinant of the other 2x2 matrix: \[ \text{det}\begin{pmatrix} 4 & 3 \\ 0 & 0 \end{pmatrix} = 0 \] Thus: \[ \text{det}(M) = -2 \times 0 = 0 \] Thus, the determinant of the matrix is \( 0 \).