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

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.