CUET UG 2025 — Mathematics Algebra

Matrix $A = \begin{bmatrix} 2 & -3 & 4 \\ -3 & 5 & x \\ 4 & 3 & 0 \end{bmatrix}$A=​2−34​−353​4x0​​ is symmetric.

For a symmetric matrix, $a_{ij} = a_{ji}$aij​=aji​ for all elements.

Comparing position (2,3) with position (3,2):

Element at (2,3) = $x$x

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

Therefore $x = 3$x=3

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Matrix $B = \begin{bmatrix} 0 & 2 & -10 \\ -2 & z & 6 \\ y & -6 & 0 \end{bmatrix}$B=​0−2y​2z−6​−1060​​ is skew-symmetric.

For a skew-symmetric matrix, all diagonal elements equal 0.

The diagonal elements are: $0, z, 0$0,z,0

Therefore $z = 0$z=0

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For a skew-symmetric matrix, $b_{ij} = -b_{ji}$bij​=−bji​ for all elements.

Comparing position (1,3) with position (3,1):

Element at (1,3) = $-10$−10

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

$$-10 = -y$$−10=−y

Therefore $y = 10$y=10

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The values are:

$$x = 3$$x=3

$$y = 10$$y=10

$$z = 0$$z=0

Now calculating:

$$xy + yz + zx$$xy+yz+zx

$$= (3)(10) + (10)(0) + (0)(3)$$=(3)(10)+(10)(0)+(0)(3)

$$= 30 + 0 + 0$$=30+0+0

$$= 30$$=30

Therefore, the value of $(xy + yz + zx) = 30$(xy+yz+zx)=30