Video Transcript
Which of the following systems of
equations can be represented by the matrix form the two-by-two matrix zero, negative
two, three, negative four multiplied by the two-by-one matrix 𝑥, 𝑦 is equal to the
two-by-one matrix five, negative six? Option (A) negative two 𝑦 equals
negative five, and negative two 𝑦 equals five. Option (B) negative two 𝑦 equals
negative five, and three 𝑥 minus four 𝑦 equals negative six. Option (C) negative two 𝑦 equals
negative five, and three 𝑥 minus four 𝑦 equals six. Option (D) negative two 𝑦 equals
five, and three 𝑥 minus four 𝑦 equals six. Or option (E) negative two 𝑦
equals five, and three 𝑥 minus four 𝑦 equals negative six.
In this question, we are given five
systems of linear equations and we need to determine which of these systems is
represented by a given matrix equation. And there are many different ways
we could answer this question. For example, we could rewrite each
of the systems of equations as a matrix equation and then compare this to the given
matrix equation. Although this method would work, we
would need to rewrite five different systems as matrix equations. Instead, it is easier to rewrite
the given matrix equation as a system of linear equations.
We can do this by evaluating the
matrix product on the left-hand side of the equation. To do this, we recall that we need
to multiply each entry of every row of the first matrix with the corresponding
entries of the columns of the second matrix and then add the results. Let’s start by applying this
process to the first row of the first matrix. The second matrix only has a single
column, and we have zero times 𝑥 plus negative two times 𝑦. This will be the element in the
first row and first column of this matrix.
We know that our resulting matrix
will have the same number of rows as the first matrix in the product and the same
number of columns as the second matrix in the product. It will be a two-by-one matrix. Of course, we already know this,
since we are told this product must be equal to the two-by-one matrix five, negative
six. We can evaluate this expression by
noting that zero times 𝑥 is equal to zero. We obtain negative two 𝑦.
We can follow the same process for
the second row of the first matrix and the only column of the second matrix. We get three 𝑥, and we add onto
this negative four 𝑦. This means that the element in the
second row and first column of this matrix product is three 𝑥 minus four 𝑦. We are told in the question that
this matrix product must be equal to the two-by-one matrix five, negative six. This means that we can set these
two matrices to be equal.
We can then recall that for two
matrices to be equal, they must have the same dimensions and all of their
corresponding entries must be equal. Of course, we already know that the
dimensions of these matrices are the same, since they are both two-by-one
matrices. However, we can use the fact that
these two matrices are equal to equate their corresponding entries. This gives us two equations, one
for each entry. We have that negative two 𝑦 must
be equal to five and that three 𝑥 minus four 𝑦 must be equal to negative six. We can then see that this system of
equations matches the system given in option (E).
It can also be worth noting that
none of the other options can be correct. For the first three options, we can
see that the first equation is negative two 𝑦 equals negative five instead of
five. This equation is not a multiple of
negative two 𝑦 equals five, so none of these can be correct. Similarly, in option (D), we note
that three 𝑥 minus four 𝑦 equals six is not a multiple of three 𝑥 minus four 𝑦
equals negative six. Hence, the correct answer is only
option (E). The system of equations represented
by the given matrix equation is negative two 𝑦 equals five, and three 𝑥 minus four
𝑦 equals negative six.
