If $A = \begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}$, then show that $A^{-1} = \dfrac{1}{6}(A - 5I)$

If A = 4 5 \2 1 , then show that A⁻¹ = (1)/(6)(A - – Class 12 Mathematics

Miscellaneous Exercise 2(A) · Q62

A = \begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}

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Step 1: $A=\begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}$ , $|A|=4(1)-2(5)=4-10=-6\neq0$ .

Step 2: Using the $2\times2$ shortcut, $A^{-1}=\dfrac{1}{-6}\begin{bmatrix}1&-5\\-2&4\end{bmatrix}=\begin{bmatrix}-\tfrac16&\tfrac56\\\tfrac13&-\tfrac23\end{bmatrix}$ .

Step 3: Now separately compute $A-5I=\begin{bmatrix}4&5\\2&1\end{bmatrix}-5\begin{bmatrix}1&0\\0&1\end{bmatrix}=\begin{bmatrix}4-5&5\\2&1-5\end{bmatrix}=\begin{bmatrix}-1&5\\2&-4\end{bmatrix}$ .

Step 4: $\dfrac16(A-5I)=\dfrac16\begin{bmatrix}-1&5\\2&-4\end{bmatrix}=\begin{bmatrix}-\tfrac16&\tfrac56\\\tfrac13&-\tfrac23\end{bmatrix}$ . …

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Q.If $A = \begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}$ , then show that $A^{-1} = \dfrac{1}{6}(A - 5I)$

Concept understanding — Inverse of a Matrix using Adjoint

Q.If A=[4521]A = \begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}A=[42​51​], then show that A−1=16(A−5I)A^{-1} = \dfrac{1}{6}(A - 5I)A−1=61​(A−5I)

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Q.If $A = \begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}$ , then show that $A^{-1} = \dfrac{1}{6}(A - 5I)$

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