Matrices & DeterminantsmediumFree

Matrices & Determinants — JEE Maths practice question

JEE Maths question with a full step-by-step solution.

Question

Let

A = \begin{pmatrix}2 & 0 & 7\\ 0 & 1 & 0\\ 1 & -2 & 1\end{pmatrix}

and

B = \begin{pmatrix}-\alpha & 14\alpha & 7\alpha\\ 0 & 1 & 0\\ \alpha & -4\alpha & -2\alpha\end{pmatrix}

. If

AB = I

, where

I

is the identity matrix of order 3, then the trace of

B

equals:

0

\dfrac{2}{5}

correct

\dfrac{1}{5}

5

Solution

Step 1: Determine

\alpha

from the

(1,1)

entry of

AB

(AB)_{11} = 2(-\alpha) + 0(0) + 7(\alpha) = 5\alpha

For

AB = I

5\alpha = 1 \implies \alpha = \dfrac{1}{5}

. Step 2: Compute the trace

\text{tr}(B) = (-\alpha) + 1 + (-2\alpha) = 1 - 3\alpha = 1 - \frac{3}{5} = \frac{2}{5}

Answer: (2)

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