Algebra

Matrices & Determinants

Medium

core

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:

✅ Correct Option: 4

For a system of equations to have infinite solutions, the determinant of the coefficient matrix must be zero.

The coefficient matrix is:

$A = \begin{vmatrix} 1 & -3 & 5 \\ 1 & -2 & 4 \\ 2 & -7 & \lambda \end{vmatrix}$

Expanding the determinant along Row 1:

$\det(A) = 1[(-2)(\lambda) - (4)(-7)] - (-3)[(1)(\lambda) - (4)(2)] + 5[(1)(-7) - (-2)(2)]$

$= 1[-2\lambda + 28] + 3[\lambda - 8] + 5[-7 + 4]$

$= -2\lambda + 28 + 3\lambda - 24 - 15$

$= \lambda + 28 - 24 - 15$

$= \lambda - 11$

Setting the determinant equal to zero:

$\lambda - 11 = 0$

$\lambda = 11$

2025: 30 May Shift 1

Let $A = [a_{ij}]_{2 \times 4}$ and $B = [b_{ij}]_{4 \times 2}$ , then $|3AB|$ is equal to

2025: 2 June Shift 1

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

2026: 18 May Shift 2

Match the LIST-I with LIST-II

LIST-I	LIST-II
A. All positive integral powers of a symmetric matrix are	I. Unit matrix.
B. Positive odd integral powers of a skew-symmetric matrix are	II. Null matrix.
C. All the matrices which are both symmetric and skew-symmetric, are	III. Skew - Symmetric matrix.
D. All the square matrices whose each diagonal element is one and each non-diagonal element is equal to zero, are	IV. Symmetric matrix.

Choose the correct answer from the options given below:

2026: 18 May Shift 1

If a matrix has 7 elements, then it can be a

(A) Scalar matrix

(B) Symmetric matrix

(D) Row matrix

Choose the correct answer from the options given below:

2026: 19 May Shift 1

Which of the following are TRUE?

(A) The inverse of $\begin{bmatrix} 3 & -3 \\ 2 & 7 \end{bmatrix}$ is $\frac{1}{27}\begin{bmatrix} 7 & 3 \\ -2 & 3 \end{bmatrix}$ .

(B) The determinant of $\begin{bmatrix} 3 & -3 \\ 2 & 7 \end{bmatrix}$ is 29.

(C) The inverse of $\begin{bmatrix} 3 & -3 \\ 2 & 7 \end{bmatrix}$ is $\frac{1}{29}\begin{bmatrix} 7 & 3 \\ -2 & 3 \end{bmatrix}$ .

(D) The adjoint of $\begin{bmatrix} 3 & -3 \\ 2 & 7 \end{bmatrix}$ is $\begin{bmatrix} 7 & 3 \\ -2 & 3 \end{bmatrix}$ .

Choose the correct answer from the options given below:

2025: 15 May Shift 2

If A is square matrix of order 3 × 3 and |adj A| = 64, then the value of |5A| is

If the system of equations x−3y+5z=3x - 3y + 5z = 3x−3y+5z=3 x−2y+4z=4x - 2y + 4z = 4x−2y+4z=4 2x−7y+λz=52x - 7y + \lambda z = 52x−7y+λz=5 has infinite number of solutions, then the value of λ\lambdaλ is: