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Explain solution rd sharma class 12 chapter adjoint and inverse of a matrix exercise multiple choice question question 31

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option (a) \begin{bmatrix} x^{-1} &0 &0 \\ 0& y^{-1} & 0\\ 0&0 &z^{-1} \end{bmatrix}

Given -A=\begin{bmatrix} x &0 &0 \\ 0& y & 0\\ 0&0 &z \end{bmatrix}

Hint – Find the inverse of matrix A using the formula: A^{-1}=\frac{adj A}{\left | A \right |}

Solution-

\begin{aligned} &A=\left[\begin{array}{lll} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{array}\right] \\ &|A|=x(y z-0)-0+0=x y z \\ &\operatorname{adj} A=\left[\begin{array}{ccc} y z & 0 & 0 \\ 0 & x z & 0 \\ 0 & 0 & x y \end{array}\right] \\ &A^{-1}=\frac{a d j A}{|A|} \\ &=\frac{1}{x y z}\left[\begin{array}{ccc} y z & 0 & 0 \\ 0 & x z & 0 \\ 0 & 0 & x y \end{array}\right] \end{aligned}

\begin{aligned} &=\left[\begin{array}{ccc} 1 / x & 0 & 0 \\ 0 & 1 / y & 0 \\ 0 & 0 & 1 / z \end{array}\right] \\ &\mathrm{A}^{-1}=\left[\begin{array}{ccc} x^{-1} & 0 & 0 \\ 0 & y^{-1} & 0 \\ 0 & 0 & \mathrm{z}^{-1} \end{array}\right] \end{aligned}

Hence, option (a) is correct.

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