Let A be a 2 × 2 real matrix and I be the identity matrix of order 2. If the roots of the equation | A - x I | = 0 be - 1 and 3 , then the sum of the diagonal elements of the matrix A 2 is _______. [2024]

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Solution

Q.
Let $A$ be a $2 \times 2$ real matrix and $I$ be the identity matrix of order 2. If the roots of the equation $| A - x I | = 0$ be $- 1$ and $3$ , then the sum of the diagonal elements of the matrix $A^{2}$ is _______. [2024]

Ans.
(10)

Let $A = {[\begin{matrix} a & b \\ c & d \end{matrix}]}_{2 \times 2}, I = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

Roots are $- 1, 3$

Sum of roots $= - 1 + 3 = 2 = t r (A)$

$a + d = 2$ ...(i)

Product of roots $= (- 1) \times 3 = - 3 = \det (A)$

$a d - b c = - 3$ ...(ii)

$A^{2} = [\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} a^{2} + b c & a b + b d \\ a c + d c & b c + d^{2} \end{matrix}]$

$tr (A^{2}) = a^{2} + 2 b c + d^{2} = {(a + d)}^{2} - 2 a d + 2 b c$

$= 2^{2} - 2 (- 3) = 4 + 6 = 10$

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