MATH 323 Lecture 6

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Elementary Matrices

System of equations $A{\vec {x}}={\vec {b}}$ can be converted to $U{\vec {x}}={\vec {c}}$, where $U=\left(\prod _{i=k}^{1}E_{i}\right)\,A$ and ${\vec {c}}=\left(\prod _{i=k}^{1}E_{i}\right)\,{\vec {b}}$. $E_{i}$ is an elementary matrix of type 1, 2, or 3.

Properties:

• Premultiplication performs rule operations on rows
• Postmultiplication performs rule operations on columns
• $E^{-1}$ is an elementary matrix of the same type.

Row Equivalence

describes that a matrix $A$ can be converted to another matrix $B$ by applying elementary matrices: $B=\left(\prod _{i=k}^{1}E_{i}\right)\,A$

If $A$ is nonsingular, then $A$ is row equivalent to $I$: ${\begin{aligned}I&=\left(\prod _{i=k}^{1}E_{i}\right)\,A=A\,A^{-1}\\A^{-1}&=\prod _{i=k}^{1}E_{i}\end{aligned}}$

Therefore, the augmented matrix $\left[{\begin{array}{c|c}A&I\end{array}}\right]$ can be converted to $\left[{\begin{array}{c|c}I&A^{-}1\end{array}}\right]$ by performing elementary row operations to convert it to reduced row echelon form.

Example:

${\begin{aligned}\left[{\begin{array}{ccc|ccc}1&4&3&1&0&0\\-1&-2&0&0&1&0\\2&2&3&0&0&1\end{array}}\right]&\rightarrow \left[{\begin{array}{ccc|ccc}1&0&0&-{\frac {1}{2}}&-{\frac {1}{2}}&{\frac {1}{2}}\\0&1&0&{\frac {1}{4}}&-{\frac {1}{4}}&-{\frac {1}{4}}\\0&0&1&{\frac {1}{6}}&{\frac {1}{2}}&{\frac {1}{6}}\end{array}}\right]\\A&={\begin{bmatrix}1&4&3\\-1&-2&0\\2&2&3\end{bmatrix}}\\A^{-1}&={\begin{bmatrix}-{\frac {1}{2}}&-{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{4}}&-{\frac {1}{4}}&-{\frac {1}{4}}\\{\frac {1}{6}}&{\frac {1}{2}}&{\frac {1}{6}}\end{bmatrix}}\end{aligned}}$

Therefore, if we have the system of equations $A{\vec {x}}={\vec {b}}$, the solution is ${\vec {x}}=A^{-1}{\vec {b}}$.

Triangular Matrices

Suppose $A$ is a $n\times n$ square matrix.

• If $A$ is of the form ${\begin{pmatrix}x_{1}&0&\dots &0\\0&\ddots &0&\vdots \\\vdots &0&\ddots &0\\0&\dots &0&x_{n}\end{pmatrix}}$, it is diagonal.
• If $A$ is of the form ${\begin{pmatrix}x_{1}&\alpha _{2}&\dots &\beta \\0&\ddots &0&\vdots \\\vdots &0&\ddots &\gamma \\\delta &0&0&x_{n}\end{pmatrix}}$, it is upper triangular.
• If $A$ is of the form ${\begin{pmatrix}x_{1}&0&\dots &0\\\alpha &\ddots &0&\vdots \\\vdots &\beta &\ddots &0\\\gamma &\dots &\delta &x_{n}\end{pmatrix}}$, it is lower triangular.

Triangular Factorization

If a $n\times n$ matrix $A$ can be reduced to strict upper triangular form using row operations of type 3, then $A=\angle U$, where $\angle$ – unit lower triangular matrix and $U$ – strictly upper triangular matrix.

Each operation is of the form $(\#)-\ell _{ij}(\#)$, where $i$ and $j$ are row indices of $A$.

$A={\begin{bmatrix}2&4&2\\1&5&2\\4&1&9\end{bmatrix}}\longrightarrow U={\begin{bmatrix}2&4&2\\0&3&1\\0&0&8\end{bmatrix}}$

$\ell _{21}={\frac {1}{2}}$, $\ell _{31}=2$, and $\ell _{32}=-3$.

If we construct a $n\times n$ matrix by replacing elements of $I_{n}$ with corresponding $\ell$ values, we get $L={\begin{bmatrix}1&0&0\\-{\frac {1}{2}}&1&0\\2&-3&1\end{bmatrix}}$ such that $A=L\,U$ and $L=\prod _{i=k}^{1}E_{i}^{-1}$, where $E_{i}^{-1}$ is a single $\ell$ replacement in $I$ for each step.

Determinants

For a $n\times n$ square matrix $A$, there is a corresponding scalar determinant of $A$

Written in any of the following formats:

$\det(A)=\left|A\right|={\begin{vmatrix}a_{11}&\dots &a_{1n}\\\vdots &\ddots &\vdots \\a_{n1}&\dots &a_{nn}\end{vmatrix}}$

A matrix is singular iff its determinant is 0.

1. For a $1\times 1$ square matrix, $|A|=a$
2. For a $2\times 2$ square matrix, $|A|=a_{11}\,a_{22}-a_{12}\,a_{21}$ (product of main diagonal minus product of "secondary diagonal")
3. For a $3\times 3$ square matrix, $|A|=a_{11}\,{\begin{vmatrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{vmatrix}}-a_{22}\,{\begin{vmatrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{vmatrix}}+a_{13}\,{\begin{vmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{vmatrix}}$ (See MATH 251 Lecture 10 for hints and tricks)