Algebra

Matrices & Determinants

Medium

core

The system of equations $x + y - z = 1, 3x + y - 2z = 3, x - y + \lambda z = 1$ has infinite number of solutions if $\lambda$ is equal to

-5/3

any real number

✅ Correct Option: 1

For the system:

$x + y - z = 1$

$3x + y - 2z = 3$

$x - y + \lambda z = 1$

The coefficient matrix is:

$\begin{vmatrix} 1 & 1 & -1 \\ 3 & 1 & -2 \\ 1 & -1 & \lambda \end{vmatrix}$

For infinite solutions, the determinant of the coefficient matrix must equal zero.

Det $= 1(1 \cdot \lambda - (-2) \cdot (-1)) - 1(3 \cdot \lambda - (-2) \cdot 1) + (-1)(3 \cdot (-1) - 1 \cdot 1)$

Det $= 1(\lambda - 2) - 1(3\lambda + 2) - 1(-3 - 1)$

Det $= \lambda - 2 - 3\lambda - 2 + 4$

Det $= -2\lambda$

Setting the determinant equal to zero:

$-2\lambda = 0$

$\lambda = 0$

With $\lambda = 0$ , the system becomes:

$x + y - z = 1$ ... (1)

$3x + y - 2z = 3$ ... (2)

$x - y = 1$ ... (3)

From equation (3): $x = 1 + y$

Substituting in equation (1):

$(1 + y) + y - z = 1$

$2y - z = 0$

$z = 2y$

Checking equation (2):

$3(1 + y) + y - 2(2y) = 3 + 3y + y - 4y = 3$ ✓

The system is consistent. Any value can be chosen for $y$ , with corresponding $x$ and $z$ values.

Therefore, $\lambda = 0$

When $\lambda \neq 0$ , the determinant $\neq 0$ , making the system have a unique solution rather than infinite solutions.

2025: 14 May Shift 1

Let $AX = B$ be a system of three linear equations in three variables. Then the system has

(A) a unique solutions if $|A| = 0$

(B) a unique solutions if $|A| \neq 0$

(D) infinitely many solutions if $|A| = 0$ and (adj A) $B = 0$

Choose the correct answer from the options given below:

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If $\begin{vmatrix} 1 & -2 & 5 \\ 2 & \lambda & -1 \\ 0 & 4 & 2\lambda \end{vmatrix} = 86$ , the sum of the values of $\lambda$ is:

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The number of all possible matrices of order $2 \times 3$ with each entry 0 or 1 is

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Which of the following are correct?

(A) If A and B are symmetric matrices such that AB = BA, then AB is symmetric.

(B) If A and B are symmetric matrices of the same order, then (A+B) is a symmetric matrix.

(D) If A and B are symmetric matrices of the same order, then (AB+BA) is a skew symmetric matrix

Choose the correct answer from the options given below:

The system of equations x+y−z=1,3x+y−2z=3,x−y+λz=1x + y - z = 1, 3x + y - 2z = 3, x - y + \lambda z = 1x+y−z=1,3x+y−2z=3,x−y+λz=1 has infinite number of solutions if λ\lambdaλ is equal to

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

The system of equations $x + y - z = 1, 3x + y - 2z = 3, x - y + \lambda z = 1$ has infinite number of solutions if $\lambda$ is equal to