For AX=B,where A=pmatrix2&5\\5&-2pmatrix and B=pmatrix7\\3pmatrix,which statements are true? Select all that apply.

GATE 2026 ES – Question 42

Mathematics Foundation · Linear Algebra · 2 marks · Multiple select

For $AX=B$,where $A=\begin{pmatrix}2&5\\5&-2\end{pmatrix}$ and $B=\begin{pmatrix}7\\3\end{pmatrix}$,which statements are true? Select all that apply.

A is skew-symmetric.
The eigenvalues of A are real.
The system is homogeneous.
The system is singular.

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Correct answer: (B) The eigenvalues of A are real.

Explanation

$A^T=A$,so A is symmetric rather than skew-symmetric, and a real symmetric matrix has real eigenvalues (here $\pm\sqrt{29}$). The nonzero right side makes the system nonhomogeneous. Its determinant is $2(-2)-25=-29\ne0$,so it is nonsingular. Only **B** is true.

GATE 2026 ES – Question 42

Explanation

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