GATE 2024 DA Q.35: solution and answer

GATE 2024 · DAQ.35 · 1 mark · NAT

Consider the $3 \times 3$ matrix $M = 134213336$ . The determinant of $(M^{2} + 12 M)$ is ______.

See the question in the paper

Tap to open full size

Answer and solution

ANSWER

The determinant of $M^{2} + 12 M$ is $0$ .

Official answer: 0Source: Official answer key, GATE 2024 DA (IISc Bengaluru)

Factor the matrix polynomial: $M^{2} + 12 M = M (M + 12 I)$ .
The determinant is multiplicative, so $det (M^{2} + 12 M) = det (M) det (M + 12 I)$ .
Expand $det (M)$ along the first row: $1 (1 \cdot 6 - 3 \cdot 3) - 2 (3 \cdot 6 - 3 \cdot 4) + 3 (3 \cdot 3 - 1 \cdot 4)$ .
This equals $1 (- 3) - 2 (6) + 3 (5) = - 3 - 12 + 15 = 0$ .
So $M$ is singular, and $0$ is an eigenvalue of $M$ . Then $M^{2} + 12 M$ has the eigenvalue $λ^{2} + 12 λ = 0$ for $λ = 0$ , which agrees with the factor $det (M) = 0$ in step 2.
Hence $det (M^{2} + 12 M) = 0 \cdot det (M + 12 I) = 0$ . The second factor is a finite number, so it does not need to be computed.

KEY IDEA

Factor $M^{2} + 12 M = M (M + 12 I)$ . If $det (M) = 0$ , the whole product has determinant $0$ , so the second factor never has to be computed.

WATCH OUT FOR

Computing $M^{2} + 12 M$ entry by entry and then taking its determinant. This is long and error-prone. Another slip is to check only whether the trace or an individual entry is zero, instead of checking $det (M)$ .

From the GATE 2024 DA paper. Every question of this paper · Download the full paper · All 6 Eigenvalues questions, with notes

GATE 2024 DA Q.35: solution and answer

Answer and solution

More Eigenvalues questions