Question

How do you interpret the determinant of a matrix $?$

Answer 1

Determinant of a matrix:

Determinant of a matrix is the scalar quantity that can be determined by using a square matrix.

If $A$ is a matrix, then determinant of $A$ is represented by $\left|A\right|$ or $det\left(A\right)$.

The square matrix has equal number of rows and columns.

Determinant of any $\begin{array}{rcl}& & 2\times 2\end{array}$ matrix $\begin{array}{rcl}\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]& =& \left|\begin{array}{cc}a& b\\ c& d\end{array}\right|\\ & =& ad-bc\end{array}$

Determinant of any $\begin{array}{rcl}& & 3\times 3\end{array}$ matrix $\begin{array}{rcl}\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]& =& \left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|\\ & =& a(ei-fh)-b(di-fg)+c(dh-eg)\end{array}$

Hence, determinant of the matrix can be found in this way.