If A= 2 2 9 4 and I= 1 0 0 1 , then 10 A^(-1), is equal to.

Correct answer: (C) A-6I. Characteristics equation of matrix A is A-xI=0. 2 2 9 4 -x 1 0 0 1 =0 ⇒ 2-x 9 2 4-x =0 ⇒x^2-6x-10=0 ∴A^2-6A-10I=0 10A^(-1)=A-6I.

Q29 / 75 · JEE Main 2020 (08 Jan Shift 2) · Mathematics

If $A = (\begin{matrix} 2 & 2 \\ 9 & 4 \end{matrix})$ and $I = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}),$ then $10 A^{- 1}$ , is equal to.

A
$A - 4 I$
B
$6 I - A$
Correct answer
C
$A - 6 I$
D
$4 I - A$

Solution

Characteristics equation of matrix $A$ is $A - x I = 0$ .

$(\begin{matrix} 2 & 2 \\ 9 & 4 \end{matrix}) - x (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) = 0$
$\Rightarrow |\begin{matrix} 2 - x \\ 9 \end{matrix} \begin{matrix} 2 \\ 4 - x \end{matrix}| = 0 \Rightarrow x^{2} - 6 x - 10 = 0$

$∴ A^{2} - 6 A - 10 I = 0$
$10 A^{- 1} = A - 6 I$ .

Hence, the correct answer is option: 'C'.

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Solution

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If $A = (\begin{matrix} 2 & 2 \\ 9 & 4 \end{matrix})$ and $I = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}),$ then $10 A^{- 1}$ , is equal to.