Let the system of linear equations $x+2 y+z=2$, $\alpha x+3 y-z=\alpha$, $-\alpha x+y+2 z=-\alpha$ be inconsistent. Then $\alpha$ is equal to :

JEE Main · 2022 · 27 Jun 2022 · Shift 1 · Matrices

Let the system of linear equations

x + 2 y + z = 2

α x + 3 y - z = α

- α x + y + 2 z = - α

be inconsistent.

Then

α

is equal to :

(A) $\frac{5}{2}$
(B) $- \frac{5}{2}$
(C) $\frac{7}{2}$
(D) $- \frac{7}{2}$ ✓ Correct

Answer: (D)

Solution

For the system to be inconsistent, the determinant of coefficients must be zero while at least one augmented determinant is non-zero.

The coefficient determinant is:

Δ = 1 α - α 231 1 - 1 2

Expanding:

Δ = 1 (6 + 1) - 2 (2 α - α) + 1 (α + 3 α) = 7 - 2 α + 4 α = 7 + 2 α

.

For inconsistency,

Δ = 0 \Rightarrow 7 + 2 α = 0 \Rightarrow α = - \frac{7}{2}

.

Verifying: When

α = - \frac{7}{2}

, the augmented determinant

Δ_{1} = 14 + 2 α = 14 - 7 = 7 \neq = 0

, confirming inconsistency.

Therefore,

α = - \frac{7}{2}

JEE Main · 2022 · 27 Jun 2022 · Shift 1 · Single correct option

Practise more: all JEE Main PYQs on Matrices →

JEE Main · 2022 · 27 Jun 2022 · Shift 1 · Matrices

Solution

More JEE Main questions on Matrices