MATH 323 Lecture 6

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Elementary Matrices

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

Properties:

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

Row Equivalence

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

If ${\displaystyle A}$ is nonsingular, then ${\displaystyle A}$ is row equivalent to ${\displaystyle I}$: {\displaystyle {\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 ${\displaystyle \left[{\begin{array}{c|c}A&I\end{array}}\right]}$ can be converted to ${\displaystyle \left[{\begin{array}{c|c}I&A^{-}1\end{array}}\right]}$ by performing elementary row operations to convert it to reduced row echelon form.

Example:

{\displaystyle {\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 ${\displaystyle A{\vec {x}}={\vec {b}}}$, the solution is ${\displaystyle {\vec {x}}=A^{-1}{\vec {b}}}$.

Triangular Matrices

Suppose ${\displaystyle A}$ is a ${\displaystyle n\times n}$ square matrix.

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

Triangular Factorization

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

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

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

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

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

Determinants

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

Written in any of the following formats:

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

A matrix is singular iff its determinant is 0.

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