JEE Main 2017 · 8 Apr · Shift 1 · Mathematics Q65

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MathematicsSingle correct+4 / -1Question 65 of 90

Let A be any 3 × 3 invertible matrix. Then which one of the following is not always true ?

\operatorname{adj}(A)=|A|\cdot A^{-1}

\operatorname{adj}(\operatorname{adj}(A))=|A|\cdot A

\operatorname{adj}(\operatorname{adj}(A))=|A|^2\cdot(\operatorname{adj}(A))^{-1}

\operatorname{adj}(\operatorname{adj}(A))=|A|\cdot(\operatorname{adj}(A))^{-1}

Answer and worked solution

Official answer: D

For a 3 × 3 matrix, $\operatorname{adj}A=|A|A^{-1}$ and $\operatorname{adj}(\operatorname{adj}A)=|A|^{n-2}A=|A|A$ . Also $\operatorname{adj}B=|B|B^{-1}$ with $|\operatorname{adj}A|=|A|^2$ , so (3) holds. (4) has $|A|$ in place of $|A|^2$ and fails unless $|A|=1$ .

Topics:Adjoint and inverse

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JEE Main 2017 · 8 Apr · Shift 1, Mathematics question 65