Question

What do determinants tell us?

Answer 1

Properties of determinants:

The determinant is a function whose input is a square matrix and whose output is a number.

For a square matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, the determinant is

$\left|A\right|\mathbf{=}\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|\mathbf{=}\mathit{a}\mathit{d}\mathbf{-}\mathit{b}\mathit{c}$

For a square matrix $A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]$, the determinant is

$\left|A\right|\mathbf{=}\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|\mathbf{=}\mathit{a}\left(\left|\begin{array}{cc}e& f\\ h& i\end{array}\right|\right)\mathbf{-}\mathit{b}\left(\left|\begin{array}{cc}d& f\\ g& i\end{array}\right|\right)\mathbf{+}\mathit{c}\left(\left|\begin{array}{cc}d& e\\ g& h\end{array}\right|\right)$

1. The determinant tells us whether the matrix is singular or non-singular
2. If the determinant is equal to zero the matrix is singular and if the determinant is a non-zero number, the matrix is non-singular.
3. The determinant is used in finding the inverse of a matrix.

Suppose $A$ is a matrix, its inverse is calculated by ${\mathit{A}}^{\mathbf{-}\mathbf{1}}\mathbf{=}\frac{\mathbf{1}}{\left|A\right|}\mathbf{·}\mathit{a}\mathit{d}\mathit{j}\mathit{A}$.

If $\left|A\right|=0,$then the matrix does not have an inverse.