Objectives
- Matrix addition and scalar multiplication.
- Matrix multiplication
- Properties.
Chapter 2 Matrix Operations
Given a system of linear equations, say
\begin{align*}
3y+4z\amp=11\\
3x-7y+4z\amp=4\\
3x-9y+6z\amp= 6.
\end{align*}
One can write it as matrix form \(A\mathbf{x}=\mathbf{b}\text{,}\)
\begin{equation*}
\begin{pmatrix}
0 \amp 3 \amp 4 \\
3 \amp -7 \amp 4\\
3 \amp -9 \amp 6
\end{pmatrix}\begin{pmatrix}
x\\
y\\
z
\end{pmatrix}=\begin{pmatrix}
11\\
4\\
6
\end{pmatrix}
\end{equation*}
Here is the procedure to solve the linear system.
- Augmented matrix.\begin{equation*} \begin{pmatrix} 0 \amp 3 \amp 4 \amp 11 \\ 3 \amp -7 \amp 4 \amp 4 \\ 3 \amp -9 \amp 6 \amp 6 \end{pmatrix} \end{equation*}
- the reduced echelon form, using
rref.
Which can be interpret as the augmented matrix of another equivalent linear system, that is,
\begin{align*}
x \amp \amp \amp \amp \amp=1\\
\amp \amp y\amp \amp\amp=1\\
\amp\amp \amp \amp z\amp= 2.
\end{align*}
