Matrices & DeterminantsmediumFree

Properties of Matrix

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

Question

A

is a diagonal matrix of order

3\times 3

that is commutative with every square matrix of order

3\times 3

under multiplication and

\text{tr}(A) = 12

, then:

|A| = 64

correct

|A| = 16

|A| = 12

|A| = 0

Solution

Step 1: Identify the structure of

A

A diagonal matrix commutes with every square matrix of the same order if and only if it is a scalar matrix, i.e.,

A = kI

for some scalar

k

. Step 2: Determine

k

and the determinant

\text{tr}(A) = \text{tr}(kI) = 3k = 12 \implies k = 4

|A| = |4I| = 4^3 = 64

Answer: (1)

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