Matrix

Determinant

The determinant is actually super widely used for a bunch of things that I originally did not expect, which is why I dedicated its own separate page.

In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix.

What actually is the determinant?

I’ve learned it so many times, but I’ve only really learned to apply formulas. Never got a fundamental understanding of how it works.

Matrix Inverse

The inverse is of $A$ is equal to $A^{-1}=\frac{1}{\det A}adjA$ Thus, $A$ is invertible if and only if $\det A\ne 0$

For $2\times 2$ Matrix

We can just apply a simple formula for determinants and adjugates. Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\in M_{2\times 2}(\mathbb{R})$ The determinant of $A$ is $\det A=ad-bc$ and the adjugate of $A$ is $adjA=\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

See below for the more generalized definitions.

For $n\times n$ matrix

Cofactor

We need to use the Cofactor expansion to calculate the determinant. Let’s first define cofactor.

Let $A\in M_{n\times n}(\mathbb{R})$ and let $A(i,j)$ be the $(n-1)\times (n-1)$ matrix obtained from $A$ by deleting the ith row and jth column of $A$. The (i, j)-cofactor of $A$, denoted by $Cij$ , is $C_{ij}=(-1{)}^{i+j}\det A(i,j)$ so the cofactor is just a number.

Determinant

Let $A\in M_{n\times n}(\mathbb{R})$. For any $i=1,\dots ,n$, we define the determinant of $A$ as the cofactor expansion of $A$ along ANY row or column $i$ of $A$. $\det A=a_{i1}{C}_{i1}+a_{i2}{C}_{i2}+\cdots +a_{in}{C}_{in}$

Cofactor Matrix and Adjugate

The cofactor matrix of $A$ is $cofA=[C_{ij}]\in M_{n\times n}(\mathbb{R})$

The adjugate of $A$ is $adjA=[C_{ij}{]}^T\in M_{n\times n}(\mathbb{R})$

Example

1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{bmatrix}$$ We compute the determinant: $$\det(A) = 1(1 \cdot 0 - 4 \cdot 6) - 2(0 \cdot 0 - 4 \cdot 5) + 3(0 \cdot 6 - 1 \cdot 5) = 1$$ Compute the cofactor matrix of $A$: $$\text{Cof}(A) = \begin{bmatrix} ei - fh & -(di - fg) & dh - eg \\ -(bi - ch) & ai - cg & -(ah - bg) \\ bf - ce & -(af - cd) & ae - bd \end{bmatrix}$$ So we have that $$\text{Cof}(A) = \begin{bmatrix} (1 \cdot 0 - 4 \cdot 6) & -(0 \cdot 0 - 4 \cdot 5) & (0 \cdot 6 - 1 \cdot 5) \\ -(2 \cdot 0 - 3 \cdot 6) & (1 \cdot 0 - 3 \cdot 5) & -(1 \cdot 6 - 2 \cdot 5) \\ (2 \cdot 4 - 3 \cdot 1) & -(1 \cdot 4 - 3 \cdot 0) & (1 \cdot 1 - 2 \cdot 0) \end{bmatrix} = \begin{bmatrix} -24 & 20 & -5 \\ 18 & -15 & 2 \\ -4 & 5 & -1 \end{bmatrix}$$ Step 3: Transpose the cofactor matrix to get $\text{adj}(A)$

\text{adj}(A) = \text{cof}(A)^T = \begin{bmatrix} -24 & 18 & -4 \ 20 & -15 & 5 \ -5 & 2 & -1 \end{bmatrix}$$ Therefore,

-24 & 18 & -4 \\ 20 & -15 & 5 \\ -5 & 2 & -1 \end{bmatrix}$$ ### Properties of determinants Let $A,B \in M_{n\times n}(\mathbb{R})$ and $k \in \mathbb{R}$. $$\det(kA) = k^n \det A$$ $$\det(A^k) = (\det A)^k$$ $$\det(AB) = (\det A)(\det B) = (\det B)(\det A) = \det(BA)$$ $$\det(A^T) = \det(A)$$ > [!danger] Danger > > > $$\det(A+B) \neq \det(A) + \det(B)$$ ### Determinants and Area Somehow the determinant is related to the area? Also volume?? ![[attachments/Pasted image 20211124004027.png]] ### Determinants and Volume Very similar derivation as area. ![[attachments/Pasted image 20211124004000.png]]