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GATE 2024 CE (CE2) – Question 48

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors · 2 marks · Numerical answer

Consider two matrices $\mathbf{A} = \begin{bmatrix} 2 & 1 & 4 \\ 1 & 0 & 3 \end{bmatrix}$ and $\mathbf{B} = \begin{bmatrix} -1 & 0 \\ 2 & 3 \\ 1 & 4 \end{bmatrix}$.
The determinant of the matrix $\mathbf{AB}$ is __________ (in integer).

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Correct answer: 10

Explanation

The product is $\mathbf{AB} = \begin{bmatrix} 2(-1) + 1(2) + 4(1) & 0 + 3 + 16 \\ -1 + 0 + 3 & 0 + 0 + 12 \end{bmatrix} = \begin{bmatrix} 4 & 19 \\ 2 & 12 \end{bmatrix}$. Its determinant is $4 \times 12 - 19 \times 2 = 48 - 38 = 10$.