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# A is an involuntary matrix by A = $\left[ {\begin{array}{*{20}{c}} 0&1&{ - 1} \\ 4&{ - 3}&4 \\ 3&{ - 3}&4 \end{array}} \right]$ then the inverse of $\dfrac{{\text{A}}}{2}$ will beA. 2AB. $\dfrac{{{{\text{A}}^{ - 1}}}}{2}$C. $\dfrac{{\text{A}}}{2}$${{\text{A}}^{ - 1}}$D. ${{\text{A}}^2}$

Last updated date: 22nd Jul 2024
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Now, according to question A is involuntary matrix which means ${{\text{A}}^2} = {\text{I}}$ where ${\text{I}}$ is an Identity matrix which is a square matrix. All the diagonal elements of the identity matrix have value equal to 1. Except diagonal elements all other elements have value which is equal to 0. Now, using the property ${{\text{A}}^2} = {\text{I}}$, we get
${{\text{A}}^2} = {\text{I}}$ $\Rightarrow$ ${\text{AA}} = {\text{I}}$ $\Rightarrow$ ${\text{A = }}{{\text{A}}^{ - 1}}$ where ${{\text{A}}^{ - 1}}$ is the inverse of A.
Now, ${\text{AA}} = {\text{I}}$ …… (1)
$\dfrac{{\text{A}}}{2}(2{\text{A) = I}}$ , where 2A is the inverse of $\dfrac{{\text{A}}}{2}$.