Matrix Equation

Matrix Notation of a Linear System (Matrix Equation)

The matrix notation of a linear system is a compact way to represent the system using matrices and vectors. A linear system can be expressed in the form:

$$\{\begin{array}{l}{x}_1+x_2-2x_3=22x_1-3x_2+2x_3=4x_1-x_2+x_3=-1\end{array}$$

We first construct our Matrix of Coefficients as:

$$A=\left[\begin{array}{ccc}1& 1& -2\\ 2& -3& 2\\ 1& -1& 1\end{array}\right]$$

Next, we create the variable vector:

$$\vec{x}=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]$$

Finally, we form the constant vector or solution vector $\vec{b}$:

$$\vec{b}=\left[\begin{array}{c}2\\ 4\\ -1\end{array}\right]$$

We can use these components to express the linear system in matrix notation as:

$$A\vec{x}=\vec{b}$$

Existence of Solutions

The equation if consistent if and only if $\vec{b}$ is a Linear Combination of the columns of $A$ OR $\vec{b}\in \mathrm{span}(\{\overrightarrow{a_1},\overrightarrow{a_2},\overrightarrow{a_3},\dots ,\overrightarrow{a_n}\})$

Ref: Vector Span

Connection to Augmented Matrices

If $A$ consists of $n$ vectors each $\in m\times 1$

$$A=\left[\begin{array}{ccccc}\overrightarrow{a_1}& \overrightarrow{a_2}& \overrightarrow{a_3}& \dots & \overrightarrow{a_n}\end{array}\right]$$

Then $A\vec{x}=\vec{b}$ is equivalent to

$$\overrightarrow{a_1}{x}_1+\overrightarrow{a_2}{x}_2+\overrightarrow{a_3}{x}_3+\cdots +\overrightarrow{a_n}{x}_n=\vec{b}$$

and to the system with with augmented matrix

$$\left[\begin{array}{ccccccc}\overrightarrow{a_1}& \overrightarrow{a_2}& \overrightarrow{a_3}& \dots & \overrightarrow{a_n}& |& \vec{b}\end{array}\right]$$

Theorem

For a Matrix $A\in m\times n$, the following statements are equivalent:

1. For each $\vec{b}\in {\mathbb{R}}^m$, the equation $A\vec{x}=\vec{b}$ has a solution.
2. Each $\vec{b}\in {\mathbb{R}}^m$ is a Linear Combination of the columns of $A$.
3. Vector Span: $\mathrm{span}(A)={\mathbb{R}}^m$
4. $A$ has a pivot position in each row when brough to REF.

Solving Using Inverse Matrix

If $A$ is an invertible matrix, the solution to the equation $A\vec{x}=\vec{b}$ can be found using the matrix inverse:

$$\vec{x}=A^{-1}\vec{b}$$

Where $A^{-1}$ is the inverse of matrix $A$.

Derivation

$$\begin{array}{rl}A\vec{x}& =\vec{b}\\ A^{-1}(A\vec{x})& =A^{-1}\vec{b}\\ (A^{-1}A)\vec{x}& =A^{-1}\vec{b}\\ I\vec{x}& =A^{-1}\vec{b}\\ \vec{x}& =A^{-1}\vec{b}\end{array}$$