Question

For which values of xis the matrix

${\displaystyle \left[\begin{array}{ccc}3& -1+x& 2\\ 3& -1& x+2\\ x+3& -1& 2\end{array}\right]}$ non-invertible?

Options

• - 4, 0

• 2, 0

• 4, 0

• - 2, 0

Answer 1

- 4, 0

Explanation:

For the given matrix to be non-invertible,

${\displaystyle \left|\begin{array}{ccc}3& -1+x& 2\\ 3& -1& x+2\\ x+3& -1& 2\end{array}\right|}$ = 0

Applying R₁ → R₁ - R₂ and R₃ → R₃ - R₂, we get

${\displaystyle \left|\begin{array}{ccc}0& x& -x\\ 3& -1& x+2\\ x& 0& -x\end{array}\right|}$ = 0

⇒ - x (- 3x - x² - 2x) - x (x) = 0

⇒ - 3x - x² - 2x + x = 0

⇒ - x² - 4x = 0

⇒ x = 0, - 4