GATE 2024 EE – Question 11

Engineering Mathematics · Linear Algebra: Matrix Algebra · 1 mark · Multiple choice

Which one of the following matrices has an inverse?

$\begin{bmatrix}1&4&8\\0&4&2\\0.5&2&4\end{bmatrix}$
$\begin{bmatrix}1&2&3\\2&4&6\\3&2&9\end{bmatrix}$
$\begin{bmatrix}1&4&8\\0&4&2\\1&2&4\end{bmatrix}$
$\begin{bmatrix}1&4&8\\0&4&2\\3&12&24\end{bmatrix}$

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Correct answer: (C) $\begin{bmatrix}1&4&8\\0&4&2\\1&2&4\end{bmatrix}$

Explanation

A matrix has an inverse when its determinant is non-zero. In (A) row 3 $=0.5\times$ row 1, in (B) row 2 $=2\times$ row 1, and in (D) row 3 $=3\times$ row 1, so all three are singular. For (C): $\det=1(16-4)-4(0-2)+8(0-4)=12+8-32=-12\ne0$, so it is invertible.