# Determinant of a Square Matrix

## Key Questions

• Without any other information, all we can say is:

$\det \left({A}^{- 1}\right) = \frac{1}{\det \left(A\right)}$

I hope that this was helpful.

• Every SQUARE matrix $n \times n$ has a determinant.
The determinant $| A |$ of a square matrix $A$ is a number that helps you to decide:

1) What kind of solutions a system (from whose coefficients you built the square matrix $A$) can have (unique, no solutions or an infinite number of solutions);

2) If your matrix $A$, considered as an operator that produce transformations on vectors (making them bigger, flipping them, reducing them...etc.), can have an inverse (operating an inverse transformation) and what is the size of the transformation produced by $A$.

You can have a look to System of Linear Equations and Eigenvalues/Eigenvectors.

• Assuming that we have a square matrix, then the determinant of the matrix is the determinant with the same elements.

Eg if we have a $2 \times 2$ matrix:

$\boldsymbol{A} = \left(\begin{matrix}a & b \\ c & d\end{matrix}\right)$

The the associated determinant given by

$D = | \boldsymbol{A} | = | \left(a , b\right) , \left(c , d\right) | = a d - b c$

• The determinant of a matrix $A$ helps you to find the inverse matrix ${A}^{- 1}$.

You can know a few things with it :

• $A$ is invertible if and only if $D e t \left(A\right) \ne 0$.

• $D e t \left({A}^{- 1}\right) = \frac{1}{D e t \left(A\right)}$

• A^(-1) = 1/(Det(A)) * ""^t((-1)^(i+j)*M_(ij)),

where $t$ means the transpose matrix of $\left({\left(- 1\right)}^{i + j} \cdot {M}_{i j}\right)$,

where $i$ is the n° of the line, $j$ is the n° of the column of $A$,

where ${\left(- 1\right)}^{i + j}$ is the cofactor in the $i$-th row and $j$-th column of $A$,

and where ${M}_{i j}$ is the minor in the $i$-th row and $j$-th column of $A$.