CUET UG 2025 — Mathematics Algebra

When a matrix is multiplied by its inverse, the result is the Identity Matrix:

$$A \times A^{-1} = I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$A×A−1=I=​100​010​001​​

$$\begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 2 \\ x & 1 & 1 \end{bmatrix} \times \frac{1}{4}\begin{bmatrix} -2 & 0 & y \\ 5 & -2 & -1 \\ 1 & 2 & -1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$​11x​101​121​​×41​​−251​0−22​y−1−1​​=​100​010​001​​

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To find $y$y, use Row 1 of $A$A and Column 3 of $A^{-1}$A−1, which equals 0:

$$\frac{1}{4}[(1)(y) + (1)(-1) + (1)(-1)] = 0$$41​[(1)(y)+(1)(−1)+(1)(−1)]=0

$$\frac{1}{4}[y - 1 - 1] = 0$$41​[y−1−1]=0

$$\frac{1}{4}[y - 2] = 0$$41​[y−2]=0

$$y - 2 = 0$$y−2=0

$$y = 2$$y=2

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To find $x$x, use Row 3 of $A$A and Column 1 of $A^{-1}$A−1, which equals 0:

$$\frac{1}{4}[(x)(-2) + (1)(5) + (1)(1)] = 0$$41​[(x)(−2)+(1)(5)+(1)(1)]=0

$$\frac{1}{4}[-2x + 5 + 1] = 0$$41​[−2x+5+1]=0

$$\frac{1}{4}[-2x + 6] = 0$$41​[−2x+6]=0

$$-2x + 6 = 0$$−2x+6=0

$$-2x = -6$$−2x=−6

$$x = 3$$x=3

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Therefore, $x = 3$x=3 and $y = 2$y=2.