Video Transcript
Express the simultaneous equations
three 𝑥 minus 24 is negative eight 𝑦 and 𝑥 is equal to three minus 𝑦 as a matrix
equation.
We’re given a set of simultaneous
linear equations three 𝑥 minus 24 is negative eight 𝑦 and 𝑥 is equal to three
minus 𝑦, which we’re asked to express as a matrix equation. Since we have two equations in two
variables 𝑥 and 𝑦, this means our result should be a matrix equation of the form
shown, that is, a two-by-two matrix of coefficients, a two-by-one matrix of the
variables, and a two-by-one matrix of the constants, remembering that an 𝑚-by-𝑛
matrix has 𝑚 rows and 𝑛 columns. So in our case, we should have a
two-by-two matrix multiplying a two-by-one matrix equal to a two-by-one matrix.
Remember that for matrix
multiplication to work, we need an 𝑚-by-𝑛 matrix multiplying an 𝑛-by-𝑝 matrix to
equal and an 𝑚-by-𝑝 matrix. The number of columns of the first
matrix and the number of rows of the second must be equal. And our resulting matrix will have
the number of rows of the first and the number of columns of the second. And this works in our case since
our 𝑛 is equal to two, 𝑚 is two, and 𝑝 is one. And so our result will be a
two-by-one column matrix.
Before we can populate our
matrices, however, we need to make sure that our simultaneous equations have a form
where we can easily read the coefficients. What that means is we want our 𝑥’s
to be aligned and our 𝑦’s to be aligned and our constants on the right-hand
side. As we can see in our equations, for
example, in equation one we have three 𝑥 minus 24 is negative eight 𝑦. So we’ll need to move our 𝑦-term
to the left-hand side and our constant term to the right-hand side. And if we add eight 𝑦 plus 24 to
both sides, our equation one then becomes three 𝑥 plus eight 𝑦 is equal to 24.
Similarly, for our second equation
two, we need to move the 𝑦-term to the left-hand side. Adding 𝑦 to both sides gives us 𝑥
plus 𝑦 is equal to three. And so, as we can see, both our
𝑥-terms and our 𝑦-terms on the left-hand side and our constants on the right-hand
side are vertically aligned. So now making some room, we can
start to populate our matrix equation.
The coefficients of 𝑥 and 𝑦,
that’s three and eight, form the first row of our two-by-two coefficient matrix. And the constant 24 on the
right-hand side is the first element in our right-hand side column matrix. The coefficients of 𝑥 and 𝑦 in
the second equation, which are both one, form the second row of our coefficient
matrix. And the constant three on the
right-hand side of equation two is the second element in our right-hand side column
matrix. So we can see that by aligning our
variables 𝑥 and 𝑦, we can simply read off the coefficients and populate our matrix
with these.
And this matrix equation is the
full matrix representation of the set of simultaneous equations three 𝑥 plus eight
𝑦 is 24 and 𝑥 plus 𝑦 is three, or equivalently three 𝑥 minus 24 is negative
eight 𝑦 and 𝑥 is equal to three minus 𝑦. If we were to apply matrix
multiplication to the left-hand side of our matrix equation, we have three 𝑥 plus
eight 𝑦 is 24, which is our first equation recovered, and 𝑥 plus 𝑦 is equal to
three, which is our second equation.
We can generalize what we’ve done
in our two-by-two examples with the following theorem. A system of 𝑚 linear equations in
the variables 𝑥 one to 𝑥 𝑛 with coefficients 𝑎 𝑖𝑗 and constants 𝑏 𝑖, where
𝑖 takes values from one to 𝑚 and 𝑗 takes values from one to 𝑛, can be written
equivalently as the matrix equation shown, where the coefficient matrix has order 𝑚
by 𝑛, the matrix of variables has order 𝑛 by one, and the matrix of constants on
the right-hand side has order 𝑚 by one. It’s important when converting our
system of 𝑚 equations in 𝑛 variables that the orders in our matrix equation follow
this pattern. Let’s see now how this works for a
set of three equations with three unknowns. And then we’ll try the process in
reverse.
