CUET UG 2025 — Mathematics Algebra

$$(A + B)^2 = A^2 + AB + BA + B^2$$(A+B)2=A2+AB+BA+B2

Since matrix multiplication is not commutative, $AB \neq BA$AB=BA in general, so the cross terms cannot be combined.

Given $(A + B)^2 = A^2 + B^2$(A+B)2=A2+B2, this means:

$$AB + BA = O$$AB+BA=O

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$$AB = \begin{bmatrix}1 & -1 \\ 2 & -1\end{bmatrix}\begin{bmatrix}a & 1 \\ b & -1\end{bmatrix}$$AB=[12​−1−1​][ab​1−1​]

$$= \begin{bmatrix}a - b & 2 \\ 2a - b & 3\end{bmatrix}$$=[a−b2a−b​23​]

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$$BA = \begin{bmatrix}a & 1 \\ b & -1\end{bmatrix}\begin{bmatrix}1 & -1 \\ 2 & -1\end{bmatrix}$$BA=[ab​1−1​][12​−1−1​]

$$= \begin{bmatrix}a+2 & -a-1 \\ b-2 & -b+1\end{bmatrix}$$=[a+2b−2​−a−1−b+1​]

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$$AB + BA = \begin{bmatrix}2a - b + 2 & 1 - a \\ 2a - 2 & 4 - b\end{bmatrix} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}$$AB+BA=[2a−b+22a−2​1−a4−b​]=[00​00​]

From entry $(1,2)$(1,2):

$$1 - a = 0$$1−a=0

$$a = 1$$a=1

From entry $(2,2)$(2,2):

$$4 - b = 0$$4−b=0

$$b = 4$$b=4

From entry $(2,1)$(2,1):

$2(1) - 2 = 0$2(1)−2=0 ✓

From entry $(1,1)$(1,1):

$2(1) - 4 + 2 = 0$2(1)−4+2=0 ✓

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$$a = 1, \quad b = 4$$a=1,b=4