Question

How do you find the determinant of \[\left( \begin{matrix}
   1 & 4 & -2 \\
   3 & -1 & 5 \\
   7 & 0 & 2 \\
\end{matrix} \right)\]?

Answer 1

Hint: First check the given matrix is square or not. Since it is a square matrix, denote the given matrix as ‘A’. So, the determinant value of ‘A’ is denoted as $\left| A \right|$, which can be calculated using the formula $\left| A \right|=a\left( ei-hf \right)-b\left( di-gf \right)+c\left( dh-ge \right)$ with reference to the matrix $A=\left( \begin{matrix}
   a & b & c \\
   d & e & f \\
   g & h & i \\
\end{matrix} \right)$. Hence, the value of $\left| A \right|$ will be the required solution.

Complete step by step solution:
Determinant: The determinant of a matrix is a special number that can be calculated from a square matrix.
Calculating the determinant: The matrix must be square (i.e. have the same number of rows as columns) for calculating the determinant. For a $3\times 3$ square matrix $A=\left( \begin{matrix}
   a & b & c \\
   d & e & f \\
   g & h & i \\
\end{matrix} \right)$, the determinant is denoted as $\left| A \right|$ which can be calculated as $\left| A \right|=a\left( ei-hf \right)-b\left( di-gf \right)+c\left( dh-ge \right)$
Assuming the given determinant as \[A=\left( \begin{matrix}
   1 & 4 & -2 \\
   3 & -1 & 5 \\
   7 & 0 & 2 \\
\end{matrix} \right)\]
This is a square matrix of order $3\times 3$
So, by comparison we get a=1, b=4, c=$-2$, d=3, e=$-1$, f=5, g=7, h=0 and i=2
Hence the determinant of ‘A’ can be calculated as
$\begin{align}
  & \left| A \right|=1\left( \left( -1 \right)\times 2-0\times 5 \right)-4\left( 3\times 2-7\times 5 \right)+\left( -2 \right)\left( 3\times 0-7\times \left( -1 \right) \right) \\
 & =1\left( -2-0 \right)-4\left( 6-30 \right)+\left( -2 \right)\left( 0-\left( -7 \right) \right) \\
 & =1\times \left( -2 \right)-4\times \left( -24 \right)+\left( -2 \right)\left( 0+7 \right) \\
 & =-2-\left( -96 \right)+\left( -2 \right)\times 7 \\
 & =-2+96+\left( -14 \right) \\
 & =-2+96-14 \\
 & =80 \\
\end{align}$
This is the required solution of the given question.

Note:
The determinant helps us find the inverse of a matrix, tells us things about the matrix that are useful in systems of linear equations, calculus and more. The determinant can be calculated by solving any one row or column. During the calculation, the sign convention of determinant should be taken care as $\left( \begin{matrix}
   + & - & + \\
   - & + & - \\
   + & - & + \\
\end{matrix} \right)$.