Video Transcript
Express the following simultaneous equations as a matrix equation: two 𝑎 minus three 𝑏 is equal to four and negative five 𝑎 plus six 𝑏 is equal to negative seven.
In this question, we’re given a pair of simultaneous equations and we’re asked to write these simultaneous equations as a matrix equation. To do this, let’s start by recalling the standard form for a matrix equation of linear simultaneous equations. It’s the form 𝑀 times 𝑋 is equal to 𝐴, where 𝑀, 𝑋, and 𝐴 are matrices. In this case, the matrix 𝑀 will be the coefficient matrix. It’s the matrix of coefficients of the variables in our simultaneous equations. Next, our matrix 𝑋 is going to be the matrix of variables. In this case, that’s 𝑎 and 𝑏. Finally, the matrix 𝐴 is called the answer or solution matrix. It’s the constant values which answer the simultaneous equations. In this case, that’s four and negative seven.
We can use this to directly write the simultaneous equations as a matrix equation. We start by writing the matrix of coefficients, where it’s important to keep the signs of the coefficients of each of the variables. And it’s worth pointing out here each variable in our simultaneous equations will give us an extra column in our coefficient matrix and each different simultaneous equation will give us a different row. And in this case, there’s two variables in two equations, so we’ll get a two-by-two coefficient matrix. In this case, that’s the two-by-two matrix two, negative three, negative five, six.
Next, we need to multiply this by the matrix 𝑋, which is the matrix of variables. That’s 𝑎 and 𝑏. And it’s important to remember we always write this as a column matrix because we need to multiply this on the left by the coefficient matrix to generate our simultaneous equations. For example, to multiply two matrices together, we multiply the corresponding entries in the rows of the first matrix with the column of the second matrix and add these together. For the first row of the first matrix and the first column of the second matrix, we would get two 𝑎 minus three 𝑏, which is the left-hand side of the first simultaneous equation.
Before we move on to our answer matrix, there’s one more thing worth noting. We can multiply these two matrices together, since the coefficient matrix is a two-by-two matrix and the variable matrix is a two-by-one matrix. Since the number of columns of the first matrix is equal to the number of rows of the second matrix, we can multiply these two matrices together. And the result will be a two-by-one matrix. And this can help us to remember or justify that the answer matrix will also be a column matrix. Each entry of the answer matrix will just be the answers to the simultaneous equations. That’s four and negative seven.
This then gives us the following matrix equation: the two-by-two matrix two, negative three, negative five, six multiplied by the two-by-one matrix 𝑎, 𝑏 is equal to the two-by-one matrix four, negative seven.
And we can check that this is a valid matrix equation of the given simultaneous equations by evaluating the matrix multiplication. We’ve already seen what this looks like for the first row of the first matrix. So let’s instead use the second row of the first matrix. We need to multiply negative five by 𝑎, and then we need to add on to this six multiplied by 𝑏. And for the values of 𝑎 and 𝑏 to satisfy this equation, we would need this expression to be equal to negative seven. We can see this is exactly the same as the second simultaneous equation. Negative five 𝑎 plus six 𝑏 needs to be equal to negative seven.
So we could leave our answer like this. However, there’s one useful property worth mentioning. We know it doesn’t matter which order we give our simultaneous equations in. So the same should be true for our matrix equation. In particular, the order we listed the simultaneous equations in determined the order we gave the rows of our coefficient matrix and the order we gave the rows of our answer matrix. So if we switch the two rows of our coefficient matrix and the two rows of our answer matrix, we get an equivalent matrix equation. In this case, that’s the two-by-two matrix negative five, six, two, negative three multiplied by the two-by-one matrix 𝑎, 𝑏 is equal to the two-by-one matrix negative seven, four.
And it’s important to reiterate here we don’t switch the orders of our variable matrix, because switching the order of our simultaneous equations does not change the order in which the variables appear. However, we can use this idea to generate more equivalent matrix equations. For example, instead of switching the order of our simultaneous equations, we can instead switch the order that the 𝑎- and 𝑏-terms appear in each of the simultaneous equations. And this would have the effect of switching the columns of the coefficient matrix and the rows of the variable matrix. However, the answer matrix would remain unchanged.
And all of these matrix equations are equivalent, and we could use any of them as our answer. In this case, we’ll say that we can rewrite the simultaneous equations given to us in the question as the matrix equation the two-by-two matrix negative five, six, two, negative three multiplied by the two-by-one matrix 𝑎, 𝑏 is equal to the two-by-one matrix negative seven, four.
