GATE CS 2024 Set 1 Q.12: Linear Algebra

The product of all eigenvalues of the matrix $\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$ is

$-1$

$0$

$1$

$2$

Official answer and solution

Official answer: B
From: Official GATE 2024 answer key

Solution

Use the property that the product of all eigenvalues of a square matrix equals its determinant.

The product of all eigenvalues of a matrix $A$ is equal to $\det(A)$ . For the given matrix:

A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}

the rows satisfy $R_1 + R_3 = (1+7, 2+8, 3+9) = (8, 10, 12) = 2R_2$ . Since the rows are linearly dependent, $\det(A) = 0$ , and therefore the product of all eigenvalues is $0$ .

Final answer

Option (B)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2024 Computer Science and Information Technology - Set 1 (CS1)
Question: 12 of 65
Section: Computer Science and Information Technology

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