CUET UG 2025 — Mathematics Algebra

A skew-symmetric matrix satisfies $A^T = -A$AT=−A, which means:

• All diagonal elements equal $0$0
• Element at position $(i,j)$(i,j) equals negative of element at position $(j,i)$(j,i)

Given matrix:

$$A = \begin{bmatrix} a & 4 & -5 \\ d & b & -6 \\ 5 & e & c \end{bmatrix}$$A=​ad5​4be​−5−6c​​

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The diagonal elements are $a$a, $b$b, and $c$c.

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

$$a = 0$$a=0

$$b = 0$$b=0

$$c = 0$$c=0

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For positions $(1,2)$(1,2) and $(2,1)$(2,1):

Element at $(1,2)$(1,2) is $4$4

Element at $(2,1)$(2,1) is $d$d

Since $A^T = -A$AT=−A:

$$d = -4$$d=−4

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For positions $(1,3)$(1,3) and $(3,1)$(3,1):

Element at $(1,3)$(1,3) is $-5$−5

Element at $(3,1)$(3,1) is $5$5

Check: $-5 = -5$−5=−5 ✓

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For positions $(2,3)$(2,3) and $(3,2)$(3,2):

Element at $(2,3)$(2,3) is $-6$−6

Element at $(3,2)$(3,2) is $e$e

Since $A^T = -A$AT=−A:

$$e = -(-6) = 6$$e=−(−6)=6

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The sum is:

$$a + b + c + d + e = 0 + 0 + 0 + (-4) + 6 = 2$$a+b+c+d+e=0+0+0+(−4)+6=2

Therefore, the value of $a + b + c + d + e$a+b+c+d+e is equal to $2$2.