# If the matrix $\begin{bmatrix} -1 & 3 & 2 \\1 & k & -3 \\1 & 4 & 5 \end{bmatrix}$ has an inverse than the value of $k$

$\begin{array}{1 1} (1)k\; is\; any\; real \;number& (2)k=-4\\(3)k\neq -4&(4)k\neq 4\end{array}$

$A=\begin{bmatrix} -1 & 3 & 2 \\1 & k & -3 \\1 & 4 & 5 \end{bmatrix}$
Given that $A^{-1}$ exists.
$|A| \neq 0$
$| A | =\begin{vmatrix} -1 & 3 & 2 \\1 & k & -3 \\1 & 4 & 5 \end{vmatrix}$
$-1 (5 k +12)-3 -3(5+3)$
$+2(4-k) \neq 0$
$-5k-12-15-9+8-2k \neq 0$
$-7 -28 \neq 0$
$7+28 \neq 0$
$7 (k+4) \neq 0$
$k+4 \neq 0$
$k \neq -4$
Hence 3 is the correct answer.