CUET UG 2025 — Mathematics Algebra

For a skew symmetric matrix, two properties hold:

• All diagonal elements are zero: $a_{ii} = 0$aii​=0
• Elements satisfy: $a_{ij} = -a_{ji}$aij​=−aji​

Given matrix:
$$\begin{bmatrix} x - 2 & 3 & -2 \\ y & 0 & -4 \\ 2 & z & 0 \end{bmatrix}$$​x−2y2​30z​−2−40​​

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

Since all diagonal elements must be zero:

$$x - 2 = 0$$x−2=0

$$x = 2$$x=2

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

Element at (1,2) $= 3$=3
Element at (2,1) $= y$=y

These must be negatives of each other:

$$3 = -y$$3=−y

$$y = -3$$y=−3

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

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

These must be negatives of each other:

$$-4 = -z$$−4=−z

$$z = 4$$z=4

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

$$x + y + z = 3$$x+y+z=3

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