Step 1: In Which We Replace Three Linear Equations With One Matrix Equation

First of all, consider the following matrix equation:

x+2y−z2x−y+3z7x−3y−2zx+2y−z2x−y+3z7x−3y−2z size 12{ left [ matrix { x+2y - z {} ## 2x - y+3z {} ## 7x - 3y - 2z } right ]} {}== 11721172 size 12{ left [ matrix { "11" {} ## 7 {} ## 2 } right ]} {}

The matrix on the left may look like a 3×3 matrix, but it is actually a 3×1 matrix. The top element is x + 2 y - z x+2y-z (all one big number), and so on.

Remember what it means for two matrices to be equal to each other. They have to have the same dimensions (). And all the elements have to be equal to each other. So for this matrix equation to be true, all three of the following equations must be satisfied:

Look familiar? Hey, this is the three equations we started with! The point is that this one matrix equation is equivalent to those three linear equations. We can replace the original three equations with one matrix equation, and then set out to solve that.

Step 2: In Which We Replace a Simple Matrix Equation with a More Complicated One

Do the following matrix multiplication. (You will need to do this by hand—since it has variables, your calculator can’t do it for you.)

1 2 − 1 2 − 1 3 7 − 3 − 2 1 2 − 1 2 − 1 3 7 − 3 − 2 size 12{ left [ matrix { 1 {} # 2 {} # - 1 {} ## 2 {} # - 1 {} # 3 {} ## 7 {} # - 3 {} # - 2{} } right ]} {} x y z x y z size 12{ left [ matrix { x {} ## y {} ## z } right ]} {}

If you did it correctly, you should have wound up with the following 3×1 matrix:

Once again, we pause to say…hey, that looks familiar! Yes, it’s the matrix that we used in Step 1. So we can now rewrite the matrix equation from Step 1 in this way:

12−12−137−3−212−12−137−3−2 size 12{ left [ matrix { 1 {} # 2 {} # - 1 {} ## 2 {} # - 1 {} # 3 {} ## 7 {} # - 3 {} # - 2{} } right ]} {}xyzxyz size 12{ left [ matrix { x {} ## y {} ## z } right ]} {}== 11721172 size 12{ left [ matrix { "11" {} ## 7 {} ## 2 } right ]} {}

Stop for a moment and make sure you’re following all this. I have shown, in two separate steps, that this matrix equation is equivalent to the three linear equations that we started with.

But this matrix equation has a nice property that the previous one did not. The first matrix (which we called [A] a long time ago) and the third one ([B]) contain only numbers. If we refer to the middle matrix as [X] then we can write our equation more concisely:

[ A ] [ X ] = [ B ] [A][X]=[B], where [ A ] = [A]= 12−12−137−3−212−12−137−3−2 size 12{ left [ matrix { 1 {} # 2 {} # - 1 {} ## 2 {} # - 1 {} # 3 {} ## 7 {} # - 3 {} # - 2{} } right ]} {}, [ X ] = [X]= xyzxyz size 12{ left [ matrix { x {} ## y {} ## z } right ]} {}, and [ B ] = [B]= 11721172 size 12{ left [ matrix { "11" {} ## 7 {} ## 2 } right ]} {}

Most importantly, [ X ] [X] contains the three variables we want to solve for! If we can solve this equation for [ X ] [X] we will have found our three variables x x, y y, and z z.

Step 3: In Which We Solve a Matrix Equation

We have rewritten our original equations as [ A ] [ X ] = [ B ] [A][X]=[B], and redefined our original goal as “solve this matrix equation for [ X ] [X].” If these were numbers, we would divide both sides by [ A ] [A]. But these are matrices, and we have never defined a division operation for matrices. Fortunately, we can do something just as good, which is multiplying both sides by [ A ] –1 [A ] –1 . (Just as, with numbers, you can replace “dividing by 3” with “multiplying by 1 3 1 3 .”)

So we’re done! [ X ] [X], which contains exactly the variables we are looking for, has been shown to be [ A ] –1 [ B ] [A ] –1 [B]. This is why we can punch that formula into our calculator and find the answers instantly.