CUET UG 2025 — Mathematics Algebra

A matrix is skew-symmetric if $A^T = -A$AT=−A

This means:

1. All diagonal elements = 0
2. Element at position (i,j) = -Element at position (j,i)

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Given matrix: $A = \begin{bmatrix} x+z & 2 & -3 \\ x & 0 & 4 \\ 3 & x-y & 0 \end{bmatrix}$A=​x+zx3​20x−y​−340​​

The diagonal elements are: $(x+z), 0, 0$(x+z),0,0

For skew-symmetric property:

$$x + z = 0$$x+z=0

$z = -x$z=−x ... (Equation 1)

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For element (1,2) and (2,1):

Element at (1,2) = 2

Element at (2,1) = $x$x

$$2 = -x$$2=−x

$$x = -2$$x=−2

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For element (2,3) and (3,2):

Element at (2,3) = 4

Element at (3,2) = $x - y$x−y

$$4 = -(x - y)$$4=−(x−y)

$$4 = -x + y$$4=−x+y

$y = 4 + x$y=4+x ... (Equation 2)

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From $x = -2$x=−2:

$$z = -x$$z=−x

$$z = -(-2)$$z=−(−2)

$$z = 2$$z=2

$$y = 4 + x$$y=4+x

$$y = 4 + (-2)$$y=4+(−2)

$$y = 2$$y=2

Final values: $x = -2, y = 2, z = 2$x=−2,y=2,z=2

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(A) $y > z > x$y>z>x → $2 > 2 > -2$2>2>−2 → FALSE ($y$y is not greater than $z$z)

(B) $x > y$x>y → $-2 > 2$−2>2 → FALSE

(C) $x + y + z > 0$x+y+z>0 → $-2 + 2 + 2 = 2 > 0$−2+2+2=2>0 → TRUE

(D) $z > x$z>x → $2 > -2$2>−2 → TRUE

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Both statements (C) and (D) are true.