Given that the matrix with elements three 𝑥 minus three, negative three, negative 10, 𝑦 minus one is equal to the matrix with elements zero, negative three, negative 10, five 𝑦 minus five, find the values of 𝑥 and 𝑦.
The key to answering this question is that these two matrices are equal to one another. Two matrices will only be equal if two things are true. Firstly, they must be of the same order, meaning they have the same number of rows and the same number of columns as one another. Secondly, corresponding elements in the two matrices must be equal, which means elements in the same position. For example, the elements in the first row and first column of the matrix must be equal to one another.
Both of our matrices have two rows and two columns. So they’re of the same order. They’re both of order two by two. If we look at the element in the first row and second column of each matrix, we see that they’re indeed equal. They’re both equal to negative three. In the same way, the elements in the second row and first column of the two matrices are also equal. They’re both equal to negative 10.
To work out the value of 𝑥, we need to look at the elements in the first row and first column of the two matrices. In the first matrix, the element here is three 𝑥 minus three. And in the second matrix, the element is zero. If the two matrices are equal, then these two elements must be equal. So we can form an equation. Three 𝑥 minus three is equal to zero. We can now solve this equation to work out the value of 𝑥.
First, we add three to each side of this equation, giving three 𝑥 is equal to three. Then, we divide by three, giving 𝑥 is equal to one. So by equating the elements in the first row and first column of each matrix, we’ve found the value of 𝑥.
To find the value of 𝑦, we need to consider the elements in the second row and second column of the two matrices, both of which have been given as expressions in terms of this unknown letter 𝑦. By equating these two expressions, we get an equation that we can solve in order to find the value of 𝑦. 𝑦 minus one is equal to five 𝑦 minus five.
Notice that 𝑦 appears on both sides of this equation. So in order to solve for 𝑦, we first want to collect all of the 𝑦 terms on the same side. We’ll collect on the right of the equation as we have a larger number of 𝑦s on this side to begin with. We’ll begin then by subtracting 𝑦 from each side of this equation. On the left-hand side, we’re just left with negative one. And on the right-hand side, we have four 𝑦 minus five.
Next, we add five to each side of the equation, giving four on the left-hand side and leaving just four 𝑦 on the right-hand side. Finally, we divide both sides of the equation by four, giving one is equal to 𝑦 or 𝑦 equals one. By choosing these values of 𝑥 and 𝑦, we’ve ensured that corresponding elements in the two matrices are equal. And as the two matrices are of the same order, the two matrices will be equal.
Our solution then is 𝑥 equals one and 𝑦 equals one.