Question

How do you evaluate determinants?

Answer 1

Evaluating Determinant of a matrix:

The determinant is special number that can be calculated from a matrix.

Determinant of matrix can find matrix is a square matrix of order $n\times n$ means number of row and column are same.

For Example:

$A={\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]}_{2\times 2},\left|\begin{array}{c}A\end{array}\right|=ad-bc$ this is determinant of $2\times 2$ matrix

For $A={\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]}_{3\times 3},\left|\begin{array}{c}\mathrm{A}\end{array}\right|=\mathrm{a}(\mathrm{ei}-\mathrm{fh})-\mathrm{b}(\mathrm{di}-\mathrm{fg})+\mathrm{c}(\mathrm{dh}-\mathrm{ge})$

In this way we can calculate the determinant of the square matrix

For Example:

$\begin{array}{rcl}A& =& \left[\begin{array}{ccc}5& 3& 1\\ 7& 2& 3\\ 2& 1& 2\end{array}\right]\\ \left|\begin{array}{c}A\end{array}\right|& =& \left|5(2\times 2-3\times 1)-3(7\times 2-3\times 5)+1(7\times 1-2\times 2)\right|\\ & =& \left|5(4-3)-3(14-15)+1(7-4)\right|\\ & =& \left|5\times 1-3(-1)+3\right|\\ \left|A\right|& =& 11\end{array}$

Hence, In this way you can evaluate the determinant of a matrix.