Question

For what value of k, the matrix \[\left( \begin{matrix}
   2-k & 4 \\
   -5 & 1 \\
\end{matrix} \right)\] is not invertible?

Answer 1

 Hint:Find the determinant of the matrix. As it’s not invertible it is equal to zero. Consider the matrix A. Find the determinant of A and equal it to zero. From that, find the value of k.

Complete step-by-step answer:
A square matrix is said to be invertible if its inverse exists and is said to be non-invertible if its determinant is equal to zero.
\[\therefore \]A square matrix that is not invertible is called singular/ degenerate. Non-square matrices \[\left( m\times n \right)\] where \[m\ne n\] do not have an inverse. In some cases there are left inverse and right inverse.
Given the matrix \[\left( \begin{matrix}
   2-k & 4 \\
   -5 & 1 \\
\end{matrix} \right)\]
Let’s write it as \[A=\left( \begin{matrix}
   2-k & 4 \\
   -5 & 1 \\
\end{matrix} \right)\]
As the matrix is non-invertible, the determinant of A is zero.
\[\left| A \right|=0\]
\[\left| \left( \begin{matrix}
   2-k & 3 \\
   -5 & 1 \\
\end{matrix} \right) \right|=0\]
To work out the determinant, multiply A by the determinant of the matrix that is not in a row or column and then sum up.
\[\begin{align}
  & \left[ \left( 2-k \right)\left( 1 \right) \right]-\left[ \left( 3 \right)\left( -5 \right) \right] \\
 & =\left( 2-k \right)-\left( -15 \right) \\
 & =2-k+15=0 \\
 & \therefore k=17 \\
\end{align}\]
\[\therefore \]Value of k = 17.

Note:
For the case of a non-invertible matrix, we took\[\left| A \right|=0\].
If the matrix was invertible, then \[\left| A \right|\ne 0\]but\[{{A}^{-1}}\].