If $A = \begin{bmatrix} x & -3 & 4 \\ 3 & y & -5\\-4&z&0 \end{bmatrix}$ is a Skew-Symmetric matrix and $adj \ A = [a_{ij}]_{3 \times3}$, then $a_{11} + a_{22} + a_{33}$ is equal to

A matrix is skew-symmetric when it equals the negative of its transpose. For a skew-symmetric matrix, all diagonal elements must be 0, and elements are opposite across the diagonal.

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Given: $A = \begin{bmatrix} x & -3 & 4 \\ 3 & y & -5\\-4&z&0 \end{bmatrix}$

Since $A$ is skew-symmetric, all diagonal elements equal 0:

$x = 0$ and $y = 0$

For the element at position $(2,3)$ and $(3,2)$:

Element$(2,3) = -$Element$(3,2)$

$-5 = -z$

$z = 5$

The matrix becomes:

$A = \begin{bmatrix} 0 & -3 & 4 \\ 3 & 0 & -5\\-4&5&0 \end{bmatrix}$

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For diagonal elements of $adj A$, we need the cofactors of the diagonal elements of $A$.

Finding $a_{11}$, which equals cofactor $C_{11}$:

Remove row 1 and column 1:

$\begin{bmatrix} 0 & -5 \\ 5 & 0 \end{bmatrix}$

Determinant $= (0)(0) - (-5)(5)$

$= 0 + 25$

$= 25$

$C_{11} = (-1)^{1+1} \times 25 = 25$

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Finding $a_{22}$, which equals cofactor $C_{22}$:

Remove row 2 and column 2:

$\begin{bmatrix} 0 & 4 \\ -4 & 0 \end{bmatrix}$

Determinant $= (0)(0) - (4)(-4)$

$= 0 + 16$

$= 16$

$C_{22} = (-1)^{2+2} \times 16 = 16$

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Finding $a_{33}$, which equals cofactor $C_{33}$:

Remove row 3 and column 3:

$\begin{bmatrix} 0 & -3 \\ 3 & 0 \end{bmatrix}$

Determinant $= (0)(0) - (-3)(3)$

$= 0 + 9$

$= 9$

$C_{33} = (-1)^{3+3} \times 9 = 9$

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$a_{11} + a_{22} + a_{33}$

$= 25 + 16 + 9$

$= 50$

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Note:

The problem defines $adj \ A = [a_{ij}]$. So $a₁₁, a₂₂, a₃₃$ here refer to the entries of the adjugate matrix, not the entries of A itself.

By definition, the adjugate (classical adjoint) matrix is the transpose of the cofactor matrix:

$\text{adj}(A)_{ij} = C_{ji}$

For the diagonal elements specifically, this means:

$a_{11} = C_{11}, \quad a_{22} = C_{22}, \quad a_{33} = C_{33}$

(Transposing doesn't change diagonal entries.)

So when we computed $C₁₁ = 25, C₂₂ = 16, C₃₃ = 9$, we were directly finding the diagonal entries of $adj \ A$, which is exactly what the problem asked for.