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GATE DA 2024 Q35: Determinant of M² + 12M for a singular matrix

Question

Consider the 3×33 \times 3 matrix M=[123313436]\mathbf{M} = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 1 & 3 \\ 4 & 3 & 6 \end{bmatrix}.

The determinant of (M2+12M)(\mathbf{M}^2 + 12\mathbf{M}) is ______.

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Numerical answer. No negative marking.
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Answer: 0 (official GATE 2024 final answer key)

Concept used: det⁡(AB)=det⁡Adet⁡B\det(AB) = \det A \det B; a singular factor makes the product singular.

First step: M2+12M=M(M+12I)\mathbf{M}^2 + 12\mathbf{M} = \mathbf{M}(\mathbf{M} + 12I), and the third row of M\mathbf{M} is the sum of the first two, so det⁡M=0\det \mathbf{M} = 0.

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Source: official GATE 2024 DA question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key

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