Let A be a matrix such that . If and are non-negative integers, then is equal to _________ . [2026]

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Solution

Q.
Let A be a $3 \times 3$ matrix such that $A + A^{T} = O$ . If $A [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}] = [\begin{matrix} 3 \\ 3 \\ 2 \end{matrix}], A^{2} [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}] = [\begin{matrix} - 3 \\ 19 \\ - 24 \end{matrix}],$ and
$de t (a dj (2 adj (A + I))) = {(2)}^{α} \cdot {(3)}^{β} \cdot {(11)}^{γ},$ $α, β, γ$ are non-negative integers, then $α + β + γ$ is equal to _________ . [2026]

Ans.
(18)

$A + A^{T} = O$

$A [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}] = [\begin{matrix} 3 \\ 3 \\ 2 \end{matrix}] and A^{2} [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}] = [\begin{matrix} - 3 \\ 19 \\ - 24 \end{matrix}]$

$A + A^{T} = O$

$A is skew symmetric matrix$

$A = [\begin{matrix} 0 & a & b \\ - a & 0 & c \\ - b & - c & 0 \end{matrix}]$

Now ATQ

$[\begin{matrix} 0 & a & b \\ - a & 0 & c \\ - b & - c & 0 \end{matrix}] [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}] = [\begin{matrix} 3 \\ 3 \\ 2 \end{matrix}]$

$a = - 3 and c = b + 2$

$A^{2} [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}] = A [\begin{matrix} 3 \\ 3 \\ 2 \end{matrix}] = [\begin{matrix} 0 & - 3 & b \\ 3 & 0 & c \\ - b & - c & 0 \end{matrix}] [\begin{matrix} 3 \\ 3 \\ 2 \end{matrix}]$

$= [\begin{matrix} - 9 + 2 b \\ 9 + 2 c \\ - 3 b - 3 c \end{matrix}] = [\begin{matrix} - 3 \\ 19 \\ - 24 \end{matrix}]$

By comparing we get

$b = 3, c = 5$

Now $A = [\begin{matrix} 0 & - 3 & 3 \\ 3 & 0 & 5 \\ - 3 & - 5 & 0 \end{matrix}]$

Let $B = A + I = [\begin{matrix} 1 & - 3 & 3 \\ 3 & 1 & 5 \\ - 3 & - 5 & 1 \end{matrix}]$

$\det (B) = 44 = 2^{2} \times 11$

we find $D = \det (adj (2 adj (B)))$

$D = 2^{14} \cdot 11^{4}$

Compare with $2^{α} 3^{β} 11^{γ}$

$α = 14, β = 0, γ = 4$

$α + β + γ = 18$

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