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Let A be a ...

Question

Let $A$ be a $3 \times 3$ matrix such that $a_{11} = a_{33} = 2$ and all the other $a_{i j} = 1$ . Let $A^{- 1} = x A^{2} + y A + z I$ , then the value of $(x + y + z)$ where $I$ is a unit matrix of order $3$ , is-

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Solution

The correct option is A $1$
$A = ⎡ ⎢ ⎣ \begin{matrix} 2 & 1 & 1 1 & 1 & 1 1 & 1 & 2 \end{matrix} ⎤ ⎥ ⎦$
$A^{- 1} = x A^{2} + y A + z I$
$\Rightarrow x A^{3} + y A^{2} + z A - I = 0$
is characteristic equation of $A$
$∣ ∣ ∣ ∣ \begin{matrix} 2 - λ & 1 & 1 1 & 1 - λ & 1 1 & 1 & 2 - λ \end{matrix} ∣ ∣ ∣ ∣ = 0$
$λ^{2} - 5 λ^{2} + 5 λ - 1 = 0$
$\Rightarrow$ $x + y + z = 1$

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Let A be a 3×3 matrix such that a11=a33=2 and all the other aij=1. Let A−1=xA2+yA+zI, then the value of (x+y+z) where I is a unit matrix of order 3, is-

The correct option is A 1A=⎡⎢⎣211111112⎤⎥⎦A−1=xA2+yA+zI⇒xA3+yA2+zA−I=0is characteristic equation of A∣∣ ∣∣2−λ1111−λ1112−λ∣∣ ∣∣=0λ2−5λ2+5λ−1=0⇒ x+y+z=1

Let $A$ be a $3 \times 3$ matrix such that $a_{11} = a_{33} = 2$ and all the other $a_{i j} = 1$ . Let $A^{- 1} = x A^{2} + y A + z I$ , then the value of $(x + y + z)$ where $I$ is a unit matrix of order $3$ , is-