If A = bmatrix 1 & 1 & 1 \\ 1 & 0 & 2 \\ x & 1 & 1 bmatrix and A^(-1) = 1/4bmatrix -2 & 0 & y \\ 5 & -2 & -1 \\ 1 & 2 & -1 bmatrix, then values of x and y, are: Option 1: x = 3, y = 2 Option 2: x = 2, y = 3 Option 3: x = -3, y = -2 Option 4: x = 3, y = -2

Answer: x = 3, y = 2. Solution: When a matrix is multiplied by its inverse, the result is the Identity Matrix: A x A^(-1) = I = bmatrix 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 bmatrix bmatrix 1 & 1 & 1 \\ 1 & 0 & 2 \\ x & 1 & 1 bmatrix x 1/4bmatrix -2 & 0 & y \\ 5 & -2 & -1 \\ 1 & 2 & -1 bmatrix = bmatrix 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 bmatrix To find y, use Row 1 of A and Column 3 of A^(-1), which equals 0: 1/4[(1)(y) + (1)(-1) + (1)(-1)] = 0 1/4[y - 1 - 1] = 0 1/4[y - 2] = 0 y - 2 = 0 y = 2 To find x, use Row 3 of A and Column 1 of A^(-1), which equals 0: 1/4[(x)(-2) + (1)(5) + (1)(1)] = 0 1/4[-2x + 5 + 1] = 0 1/4[-2x + 6] = 0 -2x + 6 = 0 -2x = -6 x = 3 Therefore, x = 3 and y = 2.

Algebra

Matrices & Determinants

Medium

common

If $A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 2 \\ x & 1 & 1 \end{bmatrix}$ and $A^{-1} = \frac{1}{4}\begin{bmatrix} -2 & 0 & y \\ 5 & -2 & -1 \\ 1 & 2 & -1 \end{bmatrix}$ , then values of x and y, are:

$x = 3, y = 2$

$x = 2, y = 3$

$x = -3, y = -2$

$x = 3, y = -2$

✅ Correct Option: 1

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

$A \times A^{-1} = I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

$\begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 2 \\ x & 1 & 1 \end{bmatrix} \times \frac{1}{4}\begin{bmatrix} -2 & 0 & y \\ 5 & -2 & -1 \\ 1 & 2 & -1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

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

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

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

$\frac{1}{4}[y - 2] = 0$

$y - 2 = 0$

$y = 2$

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

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

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

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

$-2x + 6 = 0$

$-2x = -6$

$x = 3$

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

2026: 22 May Shift 2

The number of diagonal matrices of order 3 with elements either 1 or 2, is:

2026: 29 May Shift 2

If the matrix $A = \begin{bmatrix} -2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{bmatrix}$ is given, then which of the following statements is/are TRUE?

A. A is a square matrix

B. A is a symmetric matrix

C. A is a scalar matrix

D. A is a identity matrix

Choose the correct answer from the options given below:

2026: 25 May Shift 1

If $A = \begin{bmatrix} 1 & a \\ b & 2 \end{bmatrix}$ and $B = \begin{bmatrix} a & 1 \\ 2 & -3b \end{bmatrix}$ such that $\lvert AB\rvert = \lvert 2B\rvert$ and $ab \neq -\dfrac{2}{3}$ then:

A. $\lvert A\rvert = 2$

B. $\lvert A\rvert = 4$

C. $\lvert B\rvert = 2$

D. $\lvert B\rvert = 4$

Choose the correct answer from the options given below:

2025: 2 June Shift 1

If $A$ is a skew-symmetric matrix of order 5, then $|adjA|$ is equal to

2025: 30 May Shift 2

If the system of equations

$x - 3y + 5z = 3$

$x - 2y + 4z = 4$

$2x - 7y + \lambda z = 5$

has infinite number of solutions, then the value of $\lambda$ is:

2025: 2 June Shift 1

If A is a square matrix such that $A^2=A$ and I is the identify matrix of the same order as A then $(I + 2A)^3$ -6A is equal to

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