Linear systems in matrix notation

Summary

• Given a system of \(m\) equations in \(n\) unknowns

\[a_{1,1}x_1+a_{1,2}x_2+⋯a_{1,n}x_n=b_1\]

\[a_{2,1}x_1+a_{2,2}x_2+⋯a_{2,n}x_n=b_2\]

\[⋯\]

\[a_{m,1}x_1+a_{m,2}x_2+⋯a_{m,n}x_n=b_m\]

• The \(m×n\) matrix

\[A= \begin{bmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \ddots & \vdots \\ a_{m,1} & a_{m,2} & \cdots & a_{m,n} \end{bmatrix}\]

• is called the coefficient matrix of the system.
• Example:

\[4x_1+2x_2=8\]

\[x_1-2x_2=-3\]

• has coefficient matrix

\[A=\begin{bmatrix}4 & 2\\1 & -2\end{bmatrix}\]

• In matrix notation, the system of equations can be written as

\[Ax=b\]

• where \(x\) is an \(n×1\) column vector and \(b\) is an \(m×1\) column vector given by

\[x=\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} b=\begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{bmatrix}\]

• If \(m=n\) and \(A\) is invertible then the system has a unique solution given by

\[x=A^{-1}b\]

• Example (continued)

\[x= \frac{1}{-10} \begin{bmatrix} -2 & -2\\-1 & 4\end{bmatrix} \begin{bmatrix} 8 \\ -3\end{bmatrix} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}\]