Determinants

A general way to compute it is, If If $A$ is an $n\times n$ matrix where $n=1$, $\det (A)=a_{1,1}$ If $A$ is an $n\times n$ matrix where $n>1$, $\det (A)=a_{1,1}\det \left(A_{1,1}\right)-a_{1,2}\det \left(A_{1,2}\right)+\cdots +{\left(-1\right)}^{n+1}{a}_{1,n}\det \left(A_{1,n}\right)$ $a_{i,j}$ means the element at the $i^{\text{th}}$ row and the $j^{\text{th}}$ column. $A_{i,j}$ means the Matrix if you drop (get rid of) the $i^{\text{th}}$ row and the $j^{\text{th}}$ column.

Deriving it for a $2\times 2$ Say we have $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

$$\begin{array}{rl}\det (A)& =(a_{1,1})(\det \left(A_{1,1}\right))-(a_{1,2})(\det \left(A_{1,2}\right))\\ \det (A)& =(a)\left(d\right)-(b)\left(c\right)\\ \det (A)& =ad-bc\end{array}$$

Using a Cofactor

The determinant of a matrix $A$ can be computed down any row or column of the matrix.

For example, down the $j^{\text{th}}$ column the determinant is:

$$\det (A)=a_{1,j}\det \left(A_{1,j}\right)-a_{2,j}\det \left(A_{2,j}\right)+\cdots +{\left(-1\right)}^{n+1}{a}_{n,j}\det \left(A_{n,j}\right)$$

This would be useful for a matrix with a few 0’s. Say $A=\left[\begin{array}{cccc}5& 4& 3& 2\\ 0& 1& 2& 0\\ 0& -1& 1& 0\\ 0& 1& 1& 3\end{array}\right]$ find $\det (A)$ We will use the first column due to the 3 zeros.

$$\begin{array}{rl}\det (A)& =5C_{1,1}+0C_{2,1}+0C_{3,1}+0C_{4,1}\\ & =5\cdot (-1{)}^{1+1}\cdot \det \left(\left[\begin{array}{ccc}1& 2& 0\\ -1& 1& 0\\ 1& 1& 3\end{array}\right]\right)\\ 3^{\text{rd}}\text{ column}& \\ & =5\cdot \left(0C_{1,3}+0C_{2,3}+3C_{3,3}\right)\\ & =5\cdot \left(3\cdot (-1{)}^{3+3}\det \left(\left[\begin{array}{cc}1& 2\\ -1& 1\end{array}\right]\right)\right)\\ \text{Formula}& \\ & =5(3(1\times 1-2\times -1))\\ & =\boxed{45}\end{array}$$

Triangular Matrices

The determinant of a triangular matrix is the product of the entries on the main diagonal.

Row Operations

Replacement/Addition

Add a multiple of one row to another. This does $\boxed{\text{NOT}}$ effect the determinant. $\boxed{\det A=\det B}$

Interchange

Interchange two rows to make B. One swap means, $\boxed{\det B=-\det A}$. Two One swap means, $\boxed{\det B=\det A}$. We can continue this pattern

Scaling

Multiply a row by a non-zero scalar to make B. $\boxed{\det B=k\det A}$

Invertibility

Properties

1. $\det A=\det A^T$ (Transpose).
2. A is invertible if and only if $\det A\ne 0$.
3. $\det (AB)=\det A\cdot \det B$.
4. If A is invertible, then $\det A^{-1}=\frac{1}{\det A}$.

Geometric interpretation

Watch 3b1b

TLDR

Area of parallelogram spanned by the columns columns of an $n\times n$ matrix $A$ is $\det \left(\left[\begin{array}{cc}a& c\\ b& d\end{array}\right]\right)=ad-bc$.

The volume of the parallelepiped spanned by the columns of an $n\times n$ matrix $A$ is $|\det A|$.

Linear Transformations

If we have $T:{\mathbb{R}}^n\to {\mathbb{R}}^n$, and $S$ is a parallelogram in ${\mathbb{R}}^n$, then:

$$\text{volume}(T(S))=|\det A|\cdot \text{volume}(S)$$

This can be extended to higher dimensions.