GATE 2026 AE – Question 37

Engineering Mathematics · Linear Algebra: Vector algebra, matrix algebra, systems of linear equations, rank, eigenvalues and eigenvectors · 2 marks · Multiple choice

For the matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, if the relation $a + b = c + d$ holds and $a, b, c, d \ne 0$, then which one of the following statements about $A$ is FALSE?

$\begin{bmatrix} 1 \\ 1 \end{bmatrix}$ is an eigenvector
$\lambda = a + b$ is an eigenvalue
$\lambda = d - b$ is an eigenvalue
$\lambda = d + b$ is an eigenvalue

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Correct answer: (D) $\lambda = d + b$ is an eigenvalue

Explanation

$A\begin{bmatrix}1\\1\end{bmatrix} = \begin{bmatrix}a+b\\c+d\end{bmatrix} = (a+b)\begin{bmatrix}1\\1\end{bmatrix}$, so (A) and (B) hold. The trace is $a + d$, so the other eigenvalue is $a + d - (a + b) = d - b$, so (C) holds. Then $d + b$ is not in general an eigenvalue, so (D) is false.