MATH 323 Lecture 8

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Determinants by Elimination

Each elementary matrix operation corresponds to following rules:

1. Interchanging two rows/columns of matrix changes sign of determinant
2. Multiplying single row/column of matrix by scalar has effect of multiplying determinant by scalar
3. Adding multiple of row/column to another does not change value of determinant

If $U=E_k\dots E_{1}A$ is a triangular matrix, then $|U|=|E_k|\dots |E_1|=u_{11}\dots u_{nn}$. For example,

$\left|\begin{array}{ccc}2& 1& 3\\ 4& 2& 1\\ 6& -3& 4\end{array}\right|=\left|\begin{array}{ccc}2& 1& 3\\ 0& 0& -5\\ 0& -6& -5\end{array}\right|=(-1)\left|\begin{array}{ccc}2& 1& 3\\ 0& -6& -5\\ 0& 0& -5\end{array}\right|=(-1)(2)(-6)(-5)=-60$

Cramer's Rule

Let $A=\left(a_{ij}\right)$ be a $n\times n$ nonsingular matrix.

We define the adjoint matrix of $A$ as follows:

$\begin{array}{rl}\mathrm{a}\mathrm{d}\mathrm{j}A& =\left[\begin{array}{cccc}{A}_{11}& A_{21}& \dots & A_{n1}\\ A_{12}& A_{22}& \dots & A_{n2}\\ ⋮& ⋮& \ddots & ⋮\\ A_{1n}& A_{2n}& \dots & A_{nn}\end{array}\right]\\ & ={\left(A^{*}\right)}^T\end{array}$

Where $A^{*}$ is the matrix created by substituting the cofactor $A_{ij}$ for element $a_{ij}$.

By Lemma 2.2.1, $a_{i1}{A}_{j1}+\dots +a_{in}{A}_{jn}=\{\begin{array}{ll}\left|A\right|& i=j\\ 0& i\ne j\end{array}$, and

$A\left(\mathrm{a}\mathrm{d}\mathrm{j}A\right)=\left|A\right|I=\left[\begin{array}{ccc}\left|A\right|& \dots & 0\\ ⋮& \ddots & ⋮\\ 0& \dots & \left|A\right|\end{array}\right]$

Assuming $A$ is nonsingular, $A^{-1}=\frac{1}{\left|A\right|}\mathrm{a}\mathrm{d}\mathrm{j}A$

Example

$\begin{array}{rl}A& =\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]\\ A^{*}& =\left[\begin{array}{cc}{a}_{22}& -a_{21}\\ -a_{12}& a_{11}\end{array}\right]\\ \mathrm{a}\mathrm{d}\mathrm{j}A& ={\left(A^{*}\right)}^T=\left[\begin{array}{cc}{a}_{22}& -a_{12}\\ -a_{21}& a_{11}\end{array}\right]\\ A^{-1}& =\frac{1}{a_{11}{a}_{22}-a_{12}{a}_{21}}\left[\begin{array}{cc}{a}_{22}& -a_{12}\\ -a_{21}& a_{11}\end{array}\right]\end{array}$

Formal Theorem

Let $A$ be a $n\times n$ nonsingular matrix, and let $\overrightarrow{b}\in {\mathbb{ℝ}}^n$. Let $A_i$ be the matrix obtained by replacing the $i$^th column of $A$ by $\overrightarrow{b}$. If $\hat{x}$ is the unique solution to $A\overrightarrow{x}=\overrightarrow{b}$, then

${\hat{x}}_i=\frac{\left|A_i\right|}{\left|A\right|}$ for $i=1,\dots ,n$

Example

$\begin{array}{rl}{x}_1+2x_2+x_3& =5\\ 2x_1+2x_2+x_3& =6\\ x_1+2x_2+3x_3=9\end{array}$

$\begin{array}{rl}\left|A\right|& =\left|\begin{array}{ccc}1& 2& 1\\ 2& 2& 1\\ 1& 2& 3\end{array}\right|=-4\\ \left|A_1\right|& =\left|\begin{array}{ccc}5& 2& 1\\ 6& 2& 1\\ 9& 2& 3\end{array}\right|=-4\\ \left|A_2\right|& =\left|\begin{array}{ccc}1& 5& 1\\ 2& 6& 1\\ 1& 9& 3\end{array}\right|=-4\\ \left|A_3\right|& =\left|\begin{array}{ccc}1& 2& 5\\ 2& 2& 6\\ 1& 2& 9\end{array}\right|=-8\end{array}$

$\overrightarrow{x}=⟨\frac{-4}{-4},\frac{-4}{-4},\frac{-8}{-4}⟩=⟨1,1,2⟩$