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  • If A and B are invertible matrices, then which of the following is not correct? A. adj A = |A|. A-1 B. det (A)-1 = [det (A)]-1

If A and B are invertible matrices, then which of the following is not correct?
A. adj A = |A|. A^{-1}

B. det (A)^{-1} = [det (A)]^{-1}

C. (AB)^{-1} = B^{-1} A^{-1}
D. (A + B)^{-1} = B^{-1} + A^{-1}

Answers (1)

D)

We know, that A and B are invertible matrices

\vspace{\baselineskip} Consider (AB) B^{-1} A^{-1}\\ \vspace{\baselineskip} \Rightarrow (AB) B^{-1} A^{-1} = A(BB^{-1}) A^{-1}\\ \vspace{\baselineskip} = AIA^{-1} = (AI) A^{-1}\\ \vspace{\baselineskip} = AA^{-1} = I\\ \vspace{\baselineskip} \Rightarrow (AB)^{-1} = B^{-1} A^{-1} \ldots option (C)\\
\\ \vspace{\baselineskip} Also AA^{-1} = I\\ \vspace{\baselineskip} \Rightarrow \vert AA^{-1} \vert = \vert I \vert \\ \vspace{\baselineskip} \Rightarrow \vert A \vert \vert A^{-1} \vert = 1\\

\\ \begin{aligned} &\Rightarrow|\mathrm{A}|^{-1}=\frac{1}{|\mathrm{~A}|}\\ &\therefore \operatorname{det}(A)^{-1}=[\operatorname{det}(A)]^{-1} \ldots(B)\\ &\text { We know that } \frac{|\mathrm{A}|^{-1}}{ }=\frac{\operatorname{adj} \mathrm{A}}{|\vec{A}|}\\ &\Rightarrow \text { adj } A=|A| \cdot A^{-1} \ldots \text { option }(A)\\ &\Rightarrow(\mathrm{A}+\mathrm{B})^{-1}=\frac{1}{|\mathrm{~A}+\mathrm{B}|} \operatorname{adj}(\mathrm{A}+\mathrm{B})\\ &{\text { But }}{\mathrm{B}}^{-1}+\mathrm{A}^{-1}=\frac{1}{|\mathrm{~B}|} \operatorname{adj} \mathrm{B}+\frac{1}{|\mathrm{~A}|} \text { adj } \mathrm{A} \end{aligned}

\therefore (A + B)\textsuperscript{-1} \neq B\textsuperscript{-1} + A\textsuperscript{-1}\\

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infoexpert22

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