Let A = [ 1 2 0 1 ] and B = I + adj ( A ) + ( adj A ) 2 + ? + ( adj A ) 10 . Then, the sum of all the elements of the matrix B is: [2024]

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Solution

Q.
Let $A = [\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}]$ and $B = I + adj (A) + {(adj A)}^{2} + \dots + {(adj A)}^{10} .$ Then, the sum of all the elements of the matrix $B$ is: [2024]

1 $- 110$

2 $- 124$

3 $22$

4 $- 88$

Ans.
(4)

Given that $A = [\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}], adj (A) = [\begin{matrix} 1 & - 2 \\ 0 & 1 \end{matrix}],$ ${(adj A)}^{2} = [\begin{matrix} 1 & - 4 \\ 0 & 1 \end{matrix}], {(adj A)}^{4} = [\begin{matrix} 1 & - 8 \\ 0 & 1 \end{matrix}]$

$∴$ ${(adj A)}^{r} = [\begin{matrix} 1 & - 2 r \\ 0 & 1 \end{matrix}]$

Now, $B = I + adj (A) + {(adj A)}^{2} + \dots + {(adj A)}^{10} = [\begin{matrix} 11 & \sum_{r = 0}^{10} (- 2 r) \\ 0 & 11 \end{matrix}] = [\begin{matrix} 11 & - 110 \\ 0 & 11 \end{matrix}]$

$∴$ $Sum of elements of B = 11 + (- 110) + 0 + 11 = - 88 .$

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