CUET UG 2025 — Mathematics Algebra

For a 2×2 matrix A where each element $a_{ij}$aij​ follows the given rule:

• When $i \neq j$i=j (off-diagonal): $a_{ij} = 2$aij​=2
• When $i = j$i=j (diagonal): $a_{ij} = 0$aij​=0

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$$A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$$A=[a11​a21​​a12​a22​​]

For each element:

$a_{11}$a11​: Here $i = 1, j = 1$i=1,j=1 → $i = j$i=j → $a_{11} = 0$a11​=0

$a_{12}$a12​: Here $i = 1, j = 2$i=1,j=2 → $i \neq j$i=j → $a_{12} = 2$a12​=2

$a_{21}$a21​: Here $i = 2, j = 1$i=2,j=1 → $i \neq j$i=j → $a_{21} = 2$a21​=2

$a_{22}$a22​: Here $i = 2, j = 2$i=2,j=2 → $i = j$i=j → $a_{22} = 0$a22​=0

Therefore:

$$A = \begin{bmatrix} 0 & 2 \\ 2 & 0 \end{bmatrix}$$A=[02​20​]

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For a 2×2 matrix $$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$$[ac​bd​], determinant $= ad - bc$=ad−bc

$$\det(A) = (0)(0) - (2)(2)$$det(A)=(0)(0)−(2)(2)

$$\det(A) = 0 - 4$$det(A)=0−4

$$\det(A) = -4$$det(A)=−4

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Using the property $\det(A^2) = [\det(A)]^2$det(A2)=[det(A)]2:

$$\det(A^2) = (-4)^2$$det(A2)=(−4)2

$$\det(A^2) = 16$$det(A2)=16