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Inequality

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 find the value of $(x + y + z)$ where $I$ is a unit of matrix of order 3.

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Solution

The correct option is B $- 9$
Given, $A = ⎡ ⎢ ⎣ \begin{matrix} 2 & 1 & 1 1 & 1 & 1 1 & 1 & 2 \end{matrix} ⎤ ⎥ ⎦$
We know that every square matrix satisfies its characteristic equation.
$| A - λ I | = 0$
$\Rightarrow ∣ ∣ ∣ ∣ \begin{matrix} 2 - λ & 1 & 1 1 & 1 - λ & 1 1 & 1 & 2 - λ \end{matrix} ∣ ∣ ∣ ∣ = 0$
$\Rightarrow λ^{3} - 5 λ^{2} + 5 λ - 1 = 0$
$\Rightarrow A^{3} - 5 A^{2} + 5 A - 1 = 0$
Multiplying by $A^{- 1}$ , we get
$A^{2} - 5 A + 5 I - A^{- 1} = 0$
$\Rightarrow A^{- 1} = A^{2} - 5 A + 5 I$
Comparing with given expression of $A^{- 1}$ , we get $x + y + z = - 9$

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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 find the value of (x+y+z) where I is a unit of matrix of order 3.

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 find the value of $(x + y + z)$ where $I$ is a unit of matrix of order 3.