If $\begin{bmatrix} x - 2 & 3 & -2 \\ y & 0 & -4 \\ 2 & z & 0 \end{bmatrix}$ is a skew symmetric matrix, then the value of $x + y + z$ is

For a skew symmetric matrix, two properties hold:

• All diagonal elements are zero: $a_{ii} = 0$
• Elements satisfy: $a_{ij} = -a_{ji}$

Given matrix:

$\begin{bmatrix} x - 2 & 3 & -2 \\ y & 0 & -4 \\ 2 & z & 0 \end{bmatrix}$

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The diagonal elements are $(x-2)$, $0$, and $0$.

Since all diagonal elements must be zero:

$x - 2 = 0$

$x = 2$

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For elements at position (1,2) and (2,1):

Element at (1,2) $= 3$

Element at (2,1) $= y$

These must be negatives of each other:

$3 = -y$

$y = -3$

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For elements at position (2,3) and (3,2):

Element at (2,3) $= -4$

Element at (3,2) $= z$

These must be negatives of each other:

$-4 = -z$

$z = 4$

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$x + y + z = 2 + (-3) + 4$

$x + y + z = 3$

Therefore, the value of $x + y + z$ is $3$.