Question

For what value of \[k\], the matrix \[\left[ {\begin{array}{*{20}{c}}k&2\\3&4\end{array}} \right]\] has no inverse.

Answer 1

Hint:
Here, we need to find the value of \[k\] for which the given matrix has no inverse. A matrix has no inverse if its determinant is equal to 0. We will evaluate the determinant of the given matrix, and equate it to 0 to form a linear equation in terms of \[k\]. Finally, we will solve this linear equation to obtain the required value of \[k\] for which the given matrix has no inverse.

Complete step by step solution:
A matrix is invertible only if its determinant is not equal to 0.
This means that if a matrix has no inverse, then its determinant is equal to 0.
It is given that the matrix \[\left[ {\begin{array}{*{20}{c}}k&2\\3&4\end{array}} \right]\] has no inverse.
Therefore, we get
\[\left| {\begin{array}{*{20}{c}}k&2\\3&4\end{array}} \right| = 0\]
The determinant of a square matrix \[\left| {\begin{array}{*{20}{c}}a&c\\b&d\end{array}} \right|\] with 2 rows and 2 columns is given by \[ad - bc\].
Expanding the determinant in the expression, we get
\[ \Rightarrow 4k - 6 = 0\]
This is a linear equation in terms of \[k\]. We will solve this equation to find the value of \[k\].
Adding 6 on both sides, we get
\[ \Rightarrow 4k = 6\]
Dividing both sides by 4, we get
\[ \Rightarrow k = \dfrac{6}{4}\]
Simplifying the expression, we get
\[ \Rightarrow k = \dfrac{3}{2}\]

\[\therefore \] We get the value of \[k\] as \[\dfrac{3}{2}\].

Note:
If a matrix has no inverse, then its determinant is equal to 0. A matrix whose determinant is 0 is called a singular matrix. A single matrix does not have an inverse.
We have formed a linear equation in one variable in terms of \[k\] in the solution. A linear equation in one variable is an equation of the form \[ax + b = 0\], where \[b\] and \[a\] are integers. A linear equation of the form \[ax + b = 0\] has only one solution.