ISRO CS 2023 Q.41: solution and answer

ISRO 2023 · CSQ.41 · 1 mark · MCQ

Given P is a matrix = [[3, 2, 1, 4], [4, 0, 3, 1], [6, 4, 2, 8], [2, 5, 1, 3]]. If det|P| denotes the determinant of matrix P, then which of the following is true:

(A)
det|P| is indeterminate
(B)
det|P| is negative
(C)
det |P| = 0
(D)
None of the above

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Answer and solution

ANSWER

$det P = 0$ , which is option (c).

Official answer: CSource: ISRO's revised and final answer key of 16 February 2024

Write the rows of the matrix as $R_{1} = (3, 2, 1, 4)$ , $R_{2} = (4, 0, 3, 1)$ , $R_{3} = (6, 4, 2, 8)$ , $R_{4} = (2, 5, 1, 3)$ .
Compare $R_{3}$ with $R_{1}$ : $2 R_{1} = (6, 4, 2, 8) = R_{3}$ exactly, so two rows are proportional.
Subtracting $2 R_{1}$ from $R_{3}$ is a row operation that does not change the determinant, and it produces a row of zeros: $R_{3} - 2 R_{1} = (0, 0, 0, 0)$ .
A determinant with a zero row is $0$ (expand along that row), so $det P = 0$ .
Option (a) is wrong because every square matrix has a well-defined determinant. Option (b) is wrong because $0$ is not negative. Option (d) is wrong because option (c) is true.

KEY IDEA

If one row of a square matrix is a scalar multiple of another, the rows are linearly dependent, the matrix is singular and its determinant is $0$ . Look for such a relation before expanding.

WATCH OUT FOR

Expanding the $4 \times 4$ determinant in full and making an arithmetic slip, or calling the determinant "indeterminate" when it is simply zero. A determinant always has a definite value for a square matrix.

From the ISRO CS 2023 paper. Every question of this paper · The full paper (PDF, ISRO) · All 3 ISRO Eigenvalues questions, with notes

ISRO CS 2023 Q.41: solution and answer

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