Solving a Linear System Using the Inverse Matrix

Describing the process of solving a linear system using the adjacent matrix is best done while performing an example. Suppose we have a system $A*x = B$ where $A$ is the coefficient matrix of our system, $x$ is the column vector containing our variables, and $B$ is the solution column vector. We are asked to solve for the column vector $x$ made up of variables $x_1$ , $x_2$ , and $x_3$ .

$\begin{bmatrix}1&3&3\\1&4&3\\1&3&4\end{bmatrix} \begin{bmatrix} x_1\\x_2\\x_3\end{bmatrix} = \begin{bmatrix} 12\\-10\\16\end{bmatrix}$

Typically, we would divide $B$ by $A$ to solve for $x$ , however there is no method for performing division between matrices. By taking advantage of the inverse matrix property $A^{-1}*A = 1$ , we can simply the formula to solve for the column vector $x$ . The commutative property does not apply in matrix multiplication so $A^{-1}*B \not= B*A^{-1}$ . Therefore we have have to be aware of the ‘order’ in which we multiply:

$(A^{-1}) * A * x = (A^{-1}) * B$ simplifies to $x = A^{-1}*B$

Notice that since we multiplied by $A^{-1}$ ‘first’ on the left side of the equation, we also multiply ‘first’ on the right side. Now, multiplying the inverse of matrix $A$ by matrix $B$ will yield a column vector matching our $x_1$ , $x_2$ , and $x_3$ . Below, I have used the equation $x = A^{-1}*B$ and plugged the values for $A^{-1}$ into the equation. The product between $A^{-1}$ and $B$ is shown on the far right. Note: This article assumes you know how to find the inverse of a matrix. This process is described in my article Finding The Inverse of a Matrix.

$\begin{bmatrix} x_1\\x_2\\x_3\end{bmatrix} = \begin{bmatrix}7&-3&-3\\-1&1&0\\-1&0&1\end{bmatrix} \begin{bmatrix} 12\\-10\\16\end{bmatrix} = \begin{bmatrix} 66\\-22\\4\end{bmatrix}$

Therefore, $x_1=66$ , $x_2=-22$ , and $x_3=4$ . Simple systems (i.e. this 3×3 system) are much easier to solve with algebra instead of finding the inverse of the coefficient matrix and performing matrix multiplication. This application is more practical for larger systems or while working on Matrix Theory homework.

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