GATE 2026 EC Q.26: solution and answer

GATE 2026 · ECQ.26 · 1 mark · MSQ

Consider the matrix $M = 21 - 1 13 a 10 b$ .

Which of the following options is/ are TRUE if $det (M) \neq = 0$ ?

(A)
$a = - \frac{1}{2}$ and $b = - \frac{1}{2}$
(B)
$a = \frac{1}{2}$ and $b = \frac{1}{2}$
(C)
$a = - 3$ and $b = 0$
(D)
$a = \frac{1}{2}$ and $b = - 3$

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Answer and solution

ANSWER

$det (M) = a + 5 b + 3$ . It is non-zero for (B), where it equals $6$ , and for (D), where it equals $- \frac{23}{2}$ . It is zero for (A) and (C). The answer is (B) and (D).

Official answer: B, DSource: Official answer key, GATE 2026 EC (IIT Guwahati)

Expand $det (M)$ along the third column, because it contains a zero. The cofactor signs down that column are $+, -, +$ : $det (M) = 1 \cdot 1 - 1 3 a - 0 + b \cdot 2113 .$
Evaluate the minors: $1 - 1 3 a = a - (3) (- 1) = a + 3$ and $2113 = 6 - 1 = 5$ .
Therefore $det (M) = a + 5 b + 3.$ An option is TRUE exactly when its values make $a + 5 b + 3 \neq = 0$ .
(A) $a = - \frac{1}{2}, b = - \frac{1}{2}$ : $- \frac{1}{2} - \frac{5}{2} + 3 = 0$ . $M$ is singular, so (A) is not TRUE.
(B) $a = \frac{1}{2}, b = \frac{1}{2}$ : $\frac{1}{2} + \frac{5}{2} + 3 = 6 \neq = 0$ . TRUE.
(C) $a = - 3, b = 0$ : $- 3 + 0 + 3 = 0$ . $M$ is singular, so (C) is not TRUE.
(D) $a = \frac{1}{2}, b = - 3$ : $\frac{1}{2} - 15 + 3 = - \frac{23}{2} \neq = 0$ . TRUE.
Check by expanding along the first row: $2 (3 b - 0) - 1 (b - 0) + 1 (a + 3) = a + 5 b + 3$ . Both expansions agree, so the options with $det (M) \neq = 0$ are (B) and (D).

KEY IDEA

Expand along the row or column with the most zeros to get $det (M)$ as a linear expression in $a$ and $b$ , here $a + 5 b + 3$ . Then substitute each option. Every option that makes the expression zero gives a singular matrix and must be rejected.

WATCH OUT FOR

Getting a cofactor sign wrong. In the $2 \times 2$ minor $1 - 1 3 a$ , the $- 1$ makes it $a + 3$ , not $a - 3$ . The sign pattern on the middle element also matters. Either slip changes which options give zero. For example, $a - 3 + 5 b$ would wrongly reject (B) instead of (C). Since this is an MSQ, stopping after finding one option that works also loses the mark.

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GATE 2026 EC Q.26: solution and answer

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