If matrix A=bmatrix1 & 4 & 5\\3 & 2 & 6\\0 & 1 & 0bmatrix, then Match List-I with List-II, List-I, List-II, (A) A =, (I) 1/9, (B) A^(-1) =, (II) 81, (C) adj A =, (III) 9, (D) adj (adj A) =, (IV) 6561, Choose the correct answer from the options given below: Option 1: (A) - (IV), (B) - (II), (C) - (I), (D) - (III) Option 2: (A) - (III), (B) - (I), (C) - (II), (D) - (IV) Option 3: (A) - (I), (B) - (II), (C) - (III), (D) - (IV) Option 4: (A) - (IV), (B) - (III), (C) - (II), (D) - (I)

Answer: (A) - (III), (B) - (I), (C) - (II), (D) - (IV). Solution: Expanding along the third row, A, =-1(16-53)=9, so (A)-(III). Then, A^(-1), =1/9 gives (B)-(I), adjA, =, A, ^2=81 gives (C)-(II), and, adj(adjA), =, A, ^((n-1)^2)=9^4=6561 gives (D)-(IV).

Algebra

Matrices & Determinants

Medium

applied

If matrix $A=\begin{bmatrix}1 & 4 & 5\\3 & 2 & 6\\0 & 1 & 0\end{bmatrix}$ , then

Match List-I with List-II

List-I List-II
(A) $\left\vert A \right\vert =$ (I) $\dfrac{1}{9}$
(B) $\left\vert A^{-1} \right\vert =$ (II) $81$
(C) $\left\vert \text{adj } A \right\vert =$ (III) $9$
(D) $\left\vert \text{adj } (\text{adj } A) \right\vert =$ (IV) $6561$

Choose the correct answer from the options given below:

List-I	List-II
(A) $\left\vert A \right\vert =$	(I) $\dfrac{1}{9}$
(B) $\left\vert A^{-1} \right\vert =$	(II) $81$
(C) $\left\vert \text{adj } A \right\vert =$	(III) $9$
(D) $\left\vert \text{adj } (\text{adj } A) \right\vert =$	(IV) $6561$

(A) - (IV), (B) - (II), (C) - (I), (D) - (III)

(A) - (III), (B) - (I), (C) - (II), (D) - (IV)

(A) - (I), (B) - (II), (C) - (III), (D) - (IV)

(A) - (IV), (B) - (III), (C) - (II), (D) - (I)

✅ Correct Option: 2

Expanding along the third row, $|A|=-1(1\cdot6-5\cdot3)=9$ , so (A)-(III).

Then $|A^{-1}|=\frac{1}{9}$ gives (B)-(I), $|adjA|=|A|^{2}=81$ gives (C)-(II), and $|adj(adjA)|=|A|^{(n-1)^2}=9^4=6561$ gives (D)-(IV).

2025: 27 May Shift 1

Match List-I with List-II

List-I	List-II
(A) The number of possible matrices of order 3x3 with each entry 1 or 0	(I) $2^4$
(B) The number of possible matrices of order 2x3 with each entry 1 or 0	(II) $2^9$
(C) The number of possible matrices of order 2x3 with each entry 0,1,2	(III) $2^6$
(D) The number of possible matrices of order 2x2 with each entry 1 or 0	(IV) $3^6$

Choose the correct answer from the options given below:

2025: 27 May Shift 1

For the system of linear equations

$x + y + z = 5000$

$6x + 7y + 8z = 35800$

$6x + 7y - 8z = 7000$

the values of x, y and z are:

2022: 10 Aug Shift 1

If $\begin{bmatrix} 2x-1 & -3 & 6 \\ 3 & 3y-2 & 4 \\ -6 & -4 & 4z-3 \end{bmatrix}$ is skew symmetric matrix, then $xyz$ 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: 22 May Shift 2

The area (in sq. units) of the triangle whose vertices are $(0, 0)$ , $(a, 0)$ , $(0, b)$ , is equal to

2026: 18 May Shift 2

$A=\begin{bmatrix} 0 & 1 & 3 \\ 1 & 2 & x \\ 2 & 3 & 1 \end{bmatrix}$ and $2B=\begin{bmatrix} 1 & -8 & 5 \\ -1 & 6 & -3 \\ 1 & 2y & 1 \end{bmatrix}$ . If $A=B^{-1}$ then $x+y$ is equal to

Solution

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