Question

For what value of k, the matrix $\left[ {\begin{array}{*{20}{c}}
  {2 - k}&3 \\
  { - 5}&1
\end{array}} \right]$ is not invertible?

Answer 1

Hint: Here before solving this question we need to know that if a matrix is not invertible then we use the following formula $\det A = 0$ and the matrix will be called a singular matrix. Determinant can be calculated as-
$\begin{gathered}
  \det A = \left| {\begin{array}{*{20}{c}}
  a&b \\
  c&d
\end{array}} \right| \\
   = a \times d - b \times c \\
\end{gathered} $$\begin{gathered}
  \det A = \left| {\begin{array}{*{20}{c}}
  a&b \\
  c&d
\end{array}} \right| \\
   = a \times d - b \times c \\
\end{gathered} $

Complete step-by-step answer:
The given matrix is \[\left[ {\begin{array}{*{20}{c}}
  {2 - k}&3 \\
  { - 5}&1
\end{array}} \right]\]
According to this question we have,
$\left[ {\begin{array}{*{20}{c}}
  {2 - k}&3 \\
  { - 5}&1
\end{array}} \right]$
Calculating the determinant of the given matrix
$\Delta = \left| {\begin{array}{*{20}{c}}
  {2 - k}&3 \\
  { - 5}&1
\end{array}} \right|$
Applying determinant formula
$\Delta = (2 - k)1 - 3( - 5)$
Further simplifying
$\begin{gathered}
  \Delta = 2 - k + 15 \\
  \Delta = 17 - k \\
\end{gathered} $
The given matrix is not invertible if it is a singular matrix
$\begin{gathered}
  \therefore 17 - k = 0 \\
  k = 17 \\
\end{gathered} $
Hence the value of \[kis{\text{ }}17\]

Note: Here if you know the definition of invertible we can directly find the value of k. It is very simple that non-invertible means determinant equal to 0. Basics of matrices and determinants are necessary to solve problems.