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{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 $A$ :

\left|A\right|=\sum _{k=1}^{n}a_{ik}\,A_{ik}=\sum _{k=1}^{n}a_{kj}\,A_{kj}

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 $2\times 2$ matrix $A={\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{2}2\end{bmatrix}}$ , 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{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:

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

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 $1\leq k\leq n$ .

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

Proof

First case is trivial: definition of determinant

Second case: Let $A^{*}$ 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:

$\left|E\,A\right|=-\left|A\right|=\left|E\right|\,\left|A\right|$
$\left|E\,A\right|=\alpha \left|A\right|=\left|E\right|\,\left|A\right|$ , where $\alpha$ is the scalar multiple used to produce $E$
$\left|E\,A\right|=\left|A\right|=\left|E\right|\,\left|A\right|$

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