# MATH 323 Lecture 7

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## Determinants

A matrix is nonsingular iff its determinant is nonzero.

Let ${\displaystyle A}$ be a ${\displaystyle n\times n}$ matrix.

Let ${\displaystyle M_{ij}}$ be a ${\displaystyle n-1\times n-1}$ matrix obtained from ${\displaystyle A}$ by deletion of the row and column containing ${\displaystyle a_{ij}}$. The ${\displaystyle M_{ij}}$ matrices are called minors of ${\displaystyle a_{ij}}$.

Let ${\displaystyle A_{ij}=(-1)^{i+j}\,\left|M_{ij}\right|}$. This is called the cofactor of ${\displaystyle a_{ij}}$

The determinant of a ${\displaystyle n\times n}$ matrix ${\displaystyle A}$ is a scalar associated with ${\displaystyle A}$ that is defined inductively as follows:

${\displaystyle \left|A\right|={\begin{cases}a_{11}&n=1\\\sum _{j=1}^{n}a_{1j}\,A_{1j}&n>1\end{cases}}}$

### Theorem 2.1.1

The determinant can be expressed as a cofactor expansion using any row or column of ${\displaystyle A}$:

${\displaystyle \left|A\right|=\sum _{k=1}^{n}a_{ik}\,A_{ik}=\sum _{k=1}^{n}a_{kj}\,A_{kj}}$
${\displaystyle 1\leq i,j\leq n}$

It is beneficial to choose a row or column with the maximum 0 entries. This process is called inspection

### Determinant of 2 × 2 Matrix

For a ${\displaystyle 2\times 2}$ matrix ${\displaystyle A={\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{2}2\end{bmatrix}}}$, the minors are:

• ${\displaystyle M_{11}=(a_{22})}$
• ${\displaystyle M_{12}=(a_{21})}$
• ${\displaystyle M_{21}=(a_{12})}$
• ${\displaystyle M_{22}=(a_{11})}$

and the determinant could be defined as:

{\displaystyle {\begin{aligned}\left|A\right|&=a_{11}\,a_{22}-a_{12}\,a_{21}\\&=a_{11}\,\left|M_{11}\right|-a_{12}\,\left|M_{12}\right|\\&=-a_{21}\,\left|M_{21}\right|+a_{22}\,\left|M_{22}\right|\\&=a_{11}\,\left|M_{11}\right|-a_{21}\,\left|M_{21}\right|\\&=-a_{12}\,\left|M_{12}\right|+a_{22}\,\left|M_{22}\right|&=a_{11}\,A_{11}+a_{12}\,A_{12}\\&=a_{21}\,A_{21}+a_{22}\,A_{22}\end{aligned}}}

The corresponding signs for each entry are defined by the following matrix:

${\displaystyle {\begin{bmatrix}+&-\\-&+\end{bmatrix}}}$

### Theorem 2.1.2

If ${\displaystyle A}$ is a ${\displaystyle n\times n}$ matrix, then ${\displaystyle \left|A^{T}\right|=\left|A\right|}$

## Properties of Determinants

### Lemma 2.2.1

Given a ${\displaystyle n\times n}$ matrix ${\displaystyle A}$, let ${\displaystyle A_{jk}}$ be the cofactor of ${\displaystyle a_{jk}}$ for ${\displaystyle 1\leq k\leq n}$.

${\displaystyle a_{i1}\,A_{j1}+a_{i2}\,A_{j2}+\dots +A_{in}\,A_{jn}={\begin{cases}\left|A\right|&i=j\\0&i\neq j\end{cases}}}$

#### Proof

First case is trivial: definition of determinant

Second case: Let ${\displaystyle A^{*}}$ be the matrix formed by replacing the ${\displaystyle j}$th row of ${\displaystyle A}$ with the ${\displaystyle i}$th row of ${\displaystyle A}$.

${\displaystyle A_{jk}^{*}=A_{jk}}$ because the ${\displaystyle j}$th row is deleted, but by MATH 323 Theorems#Theorem 2.1.4, the determinant of a matrix with two identical rows is 0.

### Multiplication by Elementary Matrices

What happens when we take the determinant of the product of a ${\displaystyle n\times n}$ matrix ${\displaystyle A}$ and an elementary matrix ${\displaystyle E}$?

If ${\displaystyle E}$ is of type:

1. ${\displaystyle \left|E\,A\right|=-\left|A\right|=\left|E\right|\,\left|A\right|}$
2. ${\displaystyle \left|E\,A\right|=\alpha \left|A\right|=\left|E\right|\,\left|A\right|}$, where ${\displaystyle \alpha }$ is the scalar multiple used to produce ${\displaystyle E}$
3. ${\displaystyle \left|E\,A\right|=\left|A\right|=\left|E\right|\,\left|A\right|}$

In all cases, ${\displaystyle \left|E\,A\right|=\left|E\right|\,\left|A\right|}$, and coincidentally, the determinant of ${\displaystyle E}$ for all types is:

1. -1
2. α
3. 1