Let . Then the sum of all elements of the matrix is equal to : [2026]

Question

Let $A = [\begin{matrix} 1 & 1 & 1 \\ - 2 & 0 & 1 \\ 1 & 3 & 5 \end{matrix}]$ . Then the sum of all elements of the matrix $adj (adj {(2 adj A)}^{- 1}))$ is equal to : [2026]

Smart Achievers · Accepted Answer

(4)

We have, $A = [\begin{matrix} 1 & 1 & 1 \\ - 2 & 0 & 1 \\ 1 & 3 & 5 \end{matrix}]$

|A| = 1(–3) – 1(–10 – 1) + 2(–6) = –3 + 11 – 12 = –4 $\neq$ 0

Let $B = {2 adj A)}^{- 1}$

Then $adj {(adj B) = | B |}^{n - 2} \cdot B$

$= | 2 {(adj A)}^{- 1} | \cdot 2 {(adj A)}^{- 1}$ [ $∵$ n = 1]

$= \frac{2^{3}}{| adj A |} \cdot \frac{A}{| A |} = \frac{2^{4}}{| A |^{2}} \times \frac{A}{| A |} = \frac{2^{4} A}{16 | A |} = \frac{A}{| A |}$

Sum of elements of A = 12

Sum of elements of the matrix adj (adj) B) = $\frac{12}{- 4} = - 3$ .

Solution

Q.
Let $A = [\begin{matrix} 1 & 1 & 1 \\ - 2 & 0 & 1 \\ 1 & 3 & 5 \end{matrix}]$ . Then the sum of all elements of the matrix $adj (adj {(2 adj A)}^{- 1}))$ is equal to : [2026]

1 3

2 4

3 –4

4 –3

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Solution

Q. Let A=[111–201135]. Then the sum of all elements of the matrix adj(adj(2adjA)–1)) is equal to : [2026]

1 3

2 4

3 –4

4 –3

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Q.
Let $A = [\begin{matrix} 1 & 1 & 1 \\ - 2 & 0 & 1 \\ 1 & 3 & 5 \end{matrix}]$ . Then the sum of all elements of the matrix $adj (adj {(2 adj A)}^{- 1}))$ is equal to : [2026]