Express the given set of simultaneous equations as a matrix equation. Seven 𝑥 minus three 𝑦 plus six 𝑧 equals five, five 𝑥 minus two 𝑦 plus two 𝑧 equals 11, and two 𝑥 minus three 𝑦 plus eight 𝑧 equals 10.
We have three linear equations, each containing the three variables 𝑥, 𝑦, and 𝑧. All the equations are written in the same form with the 𝑥 term first followed by the 𝑦 term and then the 𝑧 term. Completing the left-hand side and just a single constant on the right. And when a set of simultaneous equations is written in this form, it’s straightforwards to rewrite it as a matrix equation.
We write the set of simultaneous equations out again. And remember, we wanted to write this set of simultaneous equations as a single matrix equation. And the entries of the matrix, which makes it a matrix equation, are going to be taken from the coefficients of the terms on the left hand-side of our equations. So for example, in the first equation, the coefficients are seven, negative three, and six. And continuing, we also have the coefficients five, negative two, and two from the second equation. And two, negative three, and eight in the third equation.
How are these coefficients arranged in our matrix? Well, in exactly the same way that they are arranged in our equations, in the way that we laid them out. All we have to do is erase from the equations anything which is not an underlined coefficient. We erase the 𝑥s, the 𝑦s, the 𝑧s, the constants’ terms on the right-hand side of the equations, and any leftover plus or equals signs. And because it’s a matrix now, we need to put some square brackets around these numbers.
So we have a matrix, but we don’t yet have an equation. For an equation, we’re going to need some unknowns. We have three unknowns — 𝑥, 𝑦, and 𝑧 — in our original set of simultaneous equations. And we put them in that order into a three by one matrix which we multiply by. And as well as having one or more variables or unknowns, an equation must also have an equal sign. And what is this left-hand side equal to? Well, we haven’t made use of these three constant terms on the right-hand side yet. And so on the right-hand side, we now have a three by one matrix with these constant terms as the entries.
So here we have our matrix equation. We have a matrix with known entries times a matrix with unknown entries equal to another matrix with known entries. And we can check that this matrix equation really does represent the set of simultaneous equations that we started with, by multiplying out the left-hand side. Multiplying these matrices, we get a three by one matrix whose entries are seven 𝑥 minus three 𝑦 plus six 𝑧, five 𝑥 minus two 𝑦 plus two 𝑧, and two 𝑥 minus three 𝑦 plus eight 𝑧. And of course, on the right-hand side, we still have the three by one matrix: five, 11, 10.
For these two matrices to be equal, their entries must be equal. And so seven 𝑥 minus three 𝑦 plus six 𝑧 must be five. Five 𝑥 minus two 𝑦 plus two 𝑧 must be 11. And two 𝑥 minus three 𝑦 plus eight 𝑧 must be 10. And these are of course the simultaneous equations that we started with.