Question

Find the value of determinant
$\left| \begin{matrix} 2 & 4 \\\ -5 & -1 \\\ \end{matrix} \right|$.

Answer 1

The given determinant is a determinant of order two. To find the value of this determinant we will first multiply diagonal elements ( $2$ and $- 1$ ) then off-diagonal elements ( $- 5$ and $4$ ) . The determinant of this matrix is equal to the product of the diagonal element minus the product of off-diagonal elements.

Complete answer:
For a square matrix, the number of rows is equal to the number of columns, say $n$ , and also it is called a square matrix of order $n$ . A square matrix of order $n$ is also called a n-rowed square matrix. If A = \left( {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}} \\\ {{a_{21}}}&{{a_{22}}} \end{array}} \right)
is a square matrix of order $2$ as there are two rows and two columns in this matrix where the elements ${a_{11}}$ and ${a_{22}}$ are called the diagonal elements and the line along which they lie is called the principal diagonal or leading diagonal of the matrix, then the expression ${a_{11}}{a_{22}} - {a_{12}}{a_{21}}$ is defined as the determinant of $A$ .
That is, $\left| A \right| =$
$\left| \begin{matrix} {a_{11}} & {a_{12}}\\\ {a_{21}} & {a_{22}} \\\ \end{matrix} \right|$.
$\Rightarrow \left| A \right| = {a_{11}}{a_{22}} - {a_{12}}{a_{21}}$
The determinant of a square matrix of order $2$ is equal to the product of the diagonal element minus the product of off-diagonal elements.
Now, let $\left| \begin{matrix} 2 & 4 \\\ -5 & -1 \\\ \end{matrix} \right|$.
Here, we have ${a_{11}} = 2$ , ${a_{12}} = 4$ , ${a_{21}} = - 5$ , ${a_{22}} = - 1$
We know, $\left| A \right| = {a_{11}}{a_{22}} - {a_{12}}{a_{21}}$
Putting the given values
$\Rightarrow \left| A \right| = \left( 2 \right)\left( { - 1} \right) - \left( 4 \right)\left( { - 5} \right)$
On solving,
$\Rightarrow \left| A \right| = \left( { - 2} \right) - \left( { - 20} \right)$
$\Rightarrow \left| A \right| = - 2 + 20$
On further simplifying,
$\Rightarrow \left| A \right| = 18$
Therefore, the value of determinant $\left| \begin{matrix} 2 & 4 \\\ -5 & -1 \\\ \end{matrix} \right|$. is $18$ .

Note:
The given matrix has $2$ rows $2$ columns i.e., the number of rows is equal to the number columns, so it is a square matrix. One important point to note is that we can find the determinant of square matrices only. So, in this case since it is a square matrix, we can find its determinant. One of the major mistakes one makes is they simply go on finding the determinant without observing if it is a square matrix or not.