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Matrices & Determinants — JEE Maths practice question

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

Question
Let AA and BB be square matrices such that AB=0AB = 0 and BB is non-singular. Then:
A∣A∣|A| must be zero but AA may be non-zero
BAA must be the zero matrixcorrect
Cnothing can be said in general about AA
Dnone of these
Solution
Step 1: Use the non-singularity of BB Since BB is non-singular, B−1B^{-1} exists. Right-multiply AB=0AB = 0 by B−1B^{-1}:
A⋅B⋅B−1=0⋅B−1  ⟹  A⋅I=0  ⟹  A=0A\cdot B\cdot B^{-1} = 0\cdot B^{-1} \implies A\cdot I = 0 \implies A = 0
 Therefore AA must be the zero matrix. Answer: (2)
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