Matrices & DeterminantsmediumFree

Properties of Matrix

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

Question
If AA is a diagonal matrix of order 3×33\times 3 that is commutative with every square matrix of order 3×33\times 3 under multiplication and tr(A)=12\text{tr}(A) = 12, then:
A∣A∣=64|A| = 64correct
B∣A∣=16|A| = 16
C∣A∣=12|A| = 12
D∣A∣=0|A| = 0
Solution
Step 1: Identify the structure of AA A diagonal matrix commutes with every square matrix of the same order if and only if it is a scalar matrix, i.e., A=kIA = kI for some scalar kk. Step 2: Determine kk and the determinant
tr(A)=tr(kI)=3k=12  ⟹  k=4\text{tr}(A) = \text{tr}(kI) = 3k = 12 \implies k = 4
∣A∣=∣4I∣=43=64|A| = |4I| = 4^3 = 64
 Answer: (1)
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