Matrix A = [1&0&k 2&1&3 k&0&1] is not invertible for :

Question

Matrix A = [1&0&k 2&1&3 k&0&1] is not invertible for : (A) k = 1 (B) k = 2 (C) k = 0 (D) All real k

Tardigrade Admin · Accepted Answer

Let A = [1&0&k 2&1&3 k&0&1] | A | = 1 (1 - 0) - 0 + k (0 - k) = 1 - k2 As we know matrix A will not be invertible if | A | = 0 ∴ |A| = 0 ⇒ 1 - k2 = 0 ⇒ k2 = 1 ⇒ k = ± 1 For, k = ± 1, | A | = 0

Q. Matrix $A = ⎣ ⎡ 12 k 010 k 31 ⎦ ⎤$ is not invertible for :

k = 1

k = 2

k = 0

All real k

Let $A = ⎣ ⎡ 12 k 010 k 31 ⎦ ⎤$
$∣ A ∣ = 1 (1 - 0) - 0 + k (0 - k) = 1 - k^{2}$
As we know matrix A will not be invertible if | A | = 0
$∴ ∣ A ∣ = 0 \Rightarrow 1 - k^{2} = 0$
$\Rightarrow k^{2} = 1 \Rightarrow k = \pm 1$
For, $k = \pm 1, ∣ A ∣ = 0$

Q. Matrix A=⎣⎡​12k​010​k31​⎦⎤​ is not invertible for :