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.