The product of the non-zero eigenvalues of the matrix bmatrix 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 bmatrix is …

Official answer: 6. Explanation: Let the given 5 5 matrix be A. Notice the structure of the rows: - Row 1 and Row 5 are identical: [1, 0, 0, 0, 1] - Row 2, Row 3, and Row 4 are identical: [0, 1, 1, 1, 0] Since there are only 2 linearly independent rows, the rank of A is 2. Hence, exactly 2 eigenvalues are non-zero, and the remaining 3 eigenvalues are 0. We can find the non-zero eigenvalues by testing eigenvectors spanned by the independent row patterns: 1. Let v1 = [1, 0, 0, 0, 1]T: A v1 = bmatrix 2 \\ 0 \\ 0 \\ 0 \\ 2 bmatrix = 2 v1 1 = 2 2. Let v2 = [0, 1, 1, 1, 0]T: A v2 = bmatrix 0 \\ 3 \\ 3 \\ 3 \\ 0 bmatrix = 3 v2 2 = 3 Thus, the non-zero eigenvalues of the matrix are 2 and 3. Their product is: 1 2 = 2 3 = 6

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GATE CSE 2014 Set 2 Q47 · NAT · 2 marks

The product of the non-zero eigenvalues of the matrix $\begin{bmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{bmatrix}$ is _________.

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Linear Algebra in GATE CS is consistently tested across fundamental properties of matrices, systems of linear equations, and spectral theory (eigenvalues and eigenvectors). Questions heavily leverage algebraic identities such as trace-sum, determinant-product relations, rank-nullity theorem, and structural properties like orthogonality of symmetric matrices and LU decomposition. Problems frequently appear as 1-mark and 2-mark conceptual MCQs/MSQs or direct numerical computation NATs.

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Engineering Mathematics · Linear Algebra

Official IIT answer key · IIT Kharagpur · Audited Aug 2026

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GATE CSE 2014 Set 2 Q47 · NAT · 2 marks

The product of the non-zero eigenvalues of the matrix $\begin{bmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{bmatrix}$ is _________.

Numerical answer

Enter a value to check

Key concept

Linear Algebra

Step-by-step verified solution

Examiner trap & misconception

Topic notes

How GATE tests Linear Algebra

Full Linear Algebra guide →

More on Linear Algebra

All 46 questions on Linear Algebra →

Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.

The product of the non-zero eigenvalues of the matrix $\begin{bmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{bmatrix}$ is _________.

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The product of the non-zero eigenvalues of the matrix $\begin{bmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{bmatrix}$ is _________.

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The product of the non-zero eigenvalues of the matrix [1000101110011100111010001]\begin{bmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{bmatrix}​10001​01110​01110​01110​10001​​ is _________.

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The product of the non-zero eigenvalues of the matrix [1000101110011100111010001]\begin{bmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{bmatrix}​10001​01110​01110​01110​10001​​ is _________.

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The product of the non-zero eigenvalues of the matrix $\begin{bmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{bmatrix}$ is _________.

The product of the non-zero eigenvalues of the matrix $\begin{bmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 1 \end{bmatrix}$ is _________.