Solving Systems of Equations Using Matrix Inverses

A system of equations can be readily solved using the concepts of the inverse matrix and matrix multiplication.  We have seen, in the chapter on simultaneous equations, how to solve two equations with two unknowns. But suppose we have three equations with three unknowns? Or four, or five?  Such situations are more common than you might suppose in the real world.  And even if you are allowed to use a calculator, it is not at all obvious how to solve such a problem in a reasonable amount of time.

Example 1:  Solve the following system of linear equations: 

$\displaystyle x+2y-z = 11 \\ 2x-y+3z = 7 \\ 7x-3y-2z = 2 $

Step 1:  Set up a matrix $A$, which is comprised of the coefficients of all the variables from each equation from the left hand side of the equal sign:

$\displaystyle [A] = \begin{bmatrix} 1&2&-1\\2&-1&3\\7&-3&-2 \end{bmatrix}$

Step 2:  Step up a matrix $B$, which consists of the constants on the right hand side of the equal sign:

$\displaystyle [B] = \begin{bmatrix} 11\\7\\2 \end{bmatrix}$

Now, in order to determine the values of $x$, $y$, and $z$, we simply multiply the inverse of $[A]$ times $[B]$. This can be done by hand, finding the inverse matrix of $[A]$, then performing the appropriate matrix multiplication with $[B]$. 

However, if you have a graphing calculator, the situation is much easier.  Using the matrix function on the calculator, first enter both matrices.  Then calculate $[A^{-1}][B]$, that is, the inverse of matrix $[A]$, multiplied by matrix $[B]$.

The calculator responds with a $3 \times 1$ matrix, which includes the solution to the system as: $x=3$, $y=5$, and $z=2$. Solving linear equations in this way is fast and easy.