MATH 323 Lecture 7

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Determinants

A matrix is nonsingular iff its determinant is nonzero.

Let $A$ be a $n\times n$ matrix.

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

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

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

$\left|A\right|=\{\begin{array}{ll}{a}_{11}& n=1\\ \sum _{j=1}^n{a}_{1j}{A}_{1j}& n>1\end{array}$

Theorem 2.1.1

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

$\left|A\right|=\sum _{k=1}^n{a}_{ik}{A}_{ik}=\sum _{k=1}^n{a}_{kj}{A}_{kj}$
$1\le i,j\le n$

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 $2\times 2$ matrix $A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{2}2\end{array}\right]$, the minors are:

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

and the determinant could be defined as:

$\begin{array}{rl}\left|A\right|& =a_{11}{a}_{22}-a_{12}{a}_{21}\\ & =a_{11}\left|M_{11}\right|-a_{12}\left|M_{12}\right|\\ & =-a_{21}\left|M_{21}\right|+a_{22}\left|M_{22}\right|\\ & =a_{11}\left|M_{11}\right|-a_{21}\left|M_{21}\right|\\ & =-a_{12}\left|M_{12}\right|+a_{22}\left|M_{22}\right|& =a_{11}{A}_{11}+a_{12}{A}_{12}\\ & =a_{21}{A}_{21}+a_{22}{A}_{22}\end{array}$

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

$\left[\begin{array}{cc}+& -\\ -& +\end{array}\right]$

Theorem 2.1.2

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

Properties of Determinants

Lemma 2.2.1

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

$a_{i1}{A}_{j1}+a_{i2}{A}_{j2}+\dots +A_{in}{A}_{jn}=\{\begin{array}{ll}\left|A\right|& i=j\\ 0& i\ne j\end{array}$

Proof

First case is trivial: definition of determinant

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

$A_{jk}^{*}=A_{jk}$ because the $j$^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 $n\times n$ matrix $A$ and an elementary matrix $E$?

If $E$ is of type:

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

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

1. -1
2. α
3. 1