Given that 𝑥 times the two-by-two
matrix negative two, zero, negative three, negative five is equal to the two-by-two
matrix 14, zero, 21, 35, find the value of 𝑥.
We’re given an equation involving
two matrices and 𝑥, and we need to determine the value of 𝑥. First, we need to notice by looking
at our equation, 𝑥 is a number. It’s not a matrix. This is because 𝑥 is given in
lowercase. Another reason we can see this is
we’re asked to find the value of 𝑥. We’re only asked to do this when 𝑥
represents a number. So in our equation, when we
multiply our matrix by 𝐴, this is scalar multiplication of a matrix. So the first thing we’ll do is
start by 𝑥 multiplied by our matrix and apply scalar multiplication to rewrite this
in a different form.
To do this, we need to recall that
when we multiply a matrix by a scalar, we multiply every single entry inside of our
matrix by our scalar. In this case, we need to multiply
all of our entries by 𝑥. By doing this, we get the
two-by-two matrix of entries 𝑥 times negative two, 𝑥 times zero, 𝑥 times negative
three, and 𝑥 times negative five. And of course, we can evaluate or
simplify all of these entries. Doing this, we get the two-by-two
matrix negative two 𝑥, zero, negative three 𝑥, negative five 𝑥.
But remember, in the question,
we’re told that this matrix is exactly equal to the two-by-two matrix 14, zero, 21,
35. So we’ve now shown that these two
matrices are equal. To find our value of 𝑥, we’re
going to need to recall what it means for two matrices to be equal. Recall that we say that two
matrices are equal if every single entry in both of our matrices are equal and they
have the same order. Of course, both of these are
two-by-two matrices. So all we need to do is check that
all of their entries are equal.
Doing this, we get a series of
equations. By equating the entries in row one
and column one, we get 14 should be equal to negative two 𝑥. By equating the entries in row one
and column two of our matrices, we get that zero should be equal to zero. By equating the entries in row two
and column one of both of our matrices, we get that 21 should be equal to negative
three 𝑥. And finally, by equating the
entries in row two and column two of our matrices, we get that 35 should be equal to
negative five 𝑥. This gives us a system of equations
we need to solve. In fact, all of these are linear
equations, so we could call this a system of linear equations.
Firstly, we can see that the second
equation is true for all values of 𝑥. Next, we can just solve all three
of the remaining linear equations. We’ll divide the first one through
by negative two, the second one through by negative three, and the third one through
by negative five. And by doing this, we can see that
all of these are solved by 𝑥 being equal to negative seven. This means we were able to show
that the value of 𝑥 should be negative seven. However, it’s also worth pointing
out we can confirm this answer either by substituting the value of 𝑥 is equal to
negative seven into our matrix or substituting 𝑥 is equal to negative seven into
our original equation. We could then verify that this
gives us the correct solution to the equation.
Therefore, given that 𝑥 times the
two-by-two matrix negative two, zero, negative three, negative five was equal to the
two-by-two matrix 14, zero, 21, 35, we were able to show that the value of 𝑥 must
be equal to negative seven.