Answer to Question #200621 in Linear Algebra for Mpopo

Question

(9.1)Determine for which value (s) of k will the matrix below be non-singular.

2-k -3

A= 2 k+1

(9.2)Determine for which value (s) of k will the matrix below be non-singular.

2 2 1

A = 3 1 3

1 3 k

Answer 1

Solution:

(9.1):

$A=\begin{bmatrix}2-k&-3\\ 2&k+1\end{bmatrix}$

A is non-singular when $|A|\ne0$

$\begin{vmatrix}2-k&-3\\ 2&k+1\end{vmatrix}\ne0 \\ \Rightarrow(2-k)(k+1)-(-3)(2)\ne0 \\ \Rightarrow-k^2+k+2+6\ne0 \\ \Rightarrow k^2-k-8\ne0$

Solving by quadratic formula,

$\Rightarrow k\ne\frac{1+\sqrt{33}}{2},\:k\ne\frac{1-\sqrt{33}}{2}$

Thus, A is non-singular for all values of k except $k=\frac{1+\sqrt{33}}{2},\:k=\frac{1-\sqrt{33}}{2}$

(9.2):

$A=\begin{bmatrix}2&2&1\\ 3&1&3\\ 1&3&k\end{bmatrix}$

A is non-singular when $|A|\ne0$

$\Rightarrow \begin{vmatrix}2&2&1\\ 3&1&3\\ 1&3&k\end{vmatrix}\ne0 \\ \Rightarrow 2(k-9)-2(3k-3)+1(9-1)\ne0 \\ \Rightarrow 2k-18-6k+6+8\ne0 \\ \Rightarrow -4k\ne4 \\\Rightarrow k\ne-1$

Thus, A is non-singular for all values of k except $k=-1$