Question

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

Answer 1

Hint: Determinant value of a 2 x 2 determinant, \[\left| \begin{matrix}
   a & b \\
   c & d \\
\end{matrix} \right|\], will be = \[ad-bc\]. Use this formula to find the determinant value of the given determinant by taking a = 2, b = 4, c = -5 and d = -1.

Here we have to evaluate the determinant \[\left| \begin{matrix}
   2 & 4 \\
   -5 & -1 \\
\end{matrix} \right|\] .
Before proceeding with the remaining solution, we must know that the determinant is a scalar value that can be computed from the elements of a square matrix.
A square matrix is a matrix with the same number of rows and columns like the given matrix is also a square matrix as it has 2 rows and 2 columns.
We know that any 2 x 2 general determinant is of the form \[\left| \begin{matrix}
   a & b \\
   c & d \\
\end{matrix} \right|\].
Also, its determinant value is \[\left( \Delta \right)=ad-bc\].
Now let us consider the determinant given in the question as,
\[A=\left| \begin{matrix}
   2 & 4 \\
   -5 & -1 \\
\end{matrix} \right|\]
By comparing it with general 2 x 2 determinant \[\left| \begin{matrix}
   a & b \\
   c & d \\
\end{matrix} \right|\], we get
\[\begin{align}
  & a=2 \\
 & b=4 \\
 & c=-5 \\
 & d=-1 \\
\end{align}\]
So, we get the value of determinant as,
\[\left( \Delta \right)=ad-bc\]
\[=2\times \left( -1 \right)-4\left( -5 \right)\]
\[=-2+20\]
\[=18\]
So, we get the determinant value of the given determinant as 18.

Note: Here students must remember that they have to multiply the diagonal elements and take the difference of them. Some students often make this mistake of writing ab – cd as formula while the correct formula as ad – bc. Here the student must take care that “a” is the element present at the first row and first column, ”b” is the element present at the first row and second column, “c” is the element present at second row and first column and “d” is the element present at second row and second column. Also, students must note that they can find the determinant value of the “square” matrix only.