Video Transcript
Which of the following systems of equations can be represented by the matrix form the two-by-two matrix 11, two, negative three, four times the two-by-one matrix 𝑥, 𝑦 is equal to the two-by-one matrix negative five, six? Is it option (A) four 𝑦 is equal to negative five plus three 𝑥 and 11𝑥 plus two 𝑦 is equal to six? Option (B) 11𝑥 plus two 𝑦 is equal to negative five and four 𝑦 is equal to six plus three 𝑥. Option (C) 11𝑦 is equal to negative five plus three 𝑥 and four 𝑥 plus two 𝑦 is equal to six. Is it option (D) four 𝑦 is equal to negative five plus two 𝑥 and 11𝑥 plus three 𝑦 is equal to six? Or is it option (E) 11𝑦 is equal to negative five plus two 𝑥 and four 𝑥 plus three 𝑦 is equal to six?
In this question, we’re given the matrix form of a system of linear equations, and we need to determine which of five given systems is represented by this matrix form. This gives us two different methods for answering this question. One way we could do this is by rewriting all five of the given systems of equations in matrix form. We would do this by writing all five of the options in standard form, find the coefficient matrix, the variable matrix, and the constant matrix. However, we will need to follow this process for all five of the given options. So instead, let’s expand the matrix form we’re given into a system of linear equations.
To do this, we need to evaluate the matrix multiplication on the left-hand side of the equation. And we could do this by recalling to multiply two matrices together, we need to multiply the corresponding entries of the rows of the first matrix with the columns of the second matrix and add the results together. And in particular, we know that the first matrix needs to have the same number of columns as the rows of the second matrix. In this case, the first matrix is a two-by-two matrix; it has two rows and two columns. And the second matrix is a two-by-one matrix; it has two rows and one column.
So, the number of columns of the first matrix is equal to the number of columns of the second matrix. And we could also conclude the resulting matrix when we multiply these two matrices together will be a two-by-one matrix, that is, a matrix with two rows and one column, which we should’ve expected because remember we know that this product should be equal to the constant matrix given in the question. We can now evaluate the product of these two matrices. We need to do this row by row. If we start with the first row, we can see we will have 11𝑥 add two 𝑦. We can do the same with the second row of the first matrix. We get negative three multiplied by 𝑥 added to four times 𝑦. This gives us the two-by-one matrix 11𝑥 plus two 𝑦 and negative three 𝑥 plus four 𝑦.
Now that we’ve rewritten the left-hand side of the given matrix equation, we can set this equal to the right-hand side of this equation. This gives us the two-by-one matrix 11𝑥 plus two 𝑦 and negative three 𝑥 plus four 𝑦 must be equal to the two-by-one matrix negative five, six. And we can recall for two matrices to be equal, they must be of the same dimension and they must also have corresponding entries equal. And we know that both of these matrices are two-by-one matrices. So for both of these matrices to be equal, their corresponding entries must be equal.
So if we set the corresponding entries equal to each other, we get 11𝑥 plus two 𝑦 must be equal to negative five and negative three 𝑥 plus four 𝑦 must be equal to six. And it’s worth noting here this is not quite equal to any of the five given options. However, we can see that option (B) contains the equation 11𝑥 plus two 𝑦 is equal to negative five. Our second equation is not quite in the same form as the answer given in option (B), so we’ll add three 𝑥 to both sides of our second equation. And if we do this, we get four 𝑦 is equal to six plus three 𝑥, which agrees with our answer in option (B).
Therefore, we were able to show the matrix equation the two-by-two matrix 11, two, negative three, four times the two-by-one matrix 𝑥, 𝑦 is equal to the two-by-one matrix negative five, six represents the system of linear equations 11𝑥 plus two 𝑦 is equal to negative five and four 𝑦 is equal to six plus three 𝑥, which is option (B).
