Video Transcript
Find the values of 𝑥 and 𝑦, given
that the matrix 10𝑥 squared plus 10, two, negative three, nine is equal to the
matrix 20, two, two 𝑦 plus nine, nine.
We need to find the values of 𝑥
and 𝑦 that makes these two matrices equal. Remember, for two matrices to be
equal, they must have the same number of rows and columns and all of the entries in
the same row and column must be equal. We can see that for the matrices
given to us in the question, both of them have two rows and two columns. So this doesn’t help us find the
values of 𝑥 or 𝑦. Let’s instead use the fact that all
entries in the same row and column must be equal.
Let’s start with the first row and
the first column. In our first matrix, this entry is
10𝑥 squared plus 10. In our second matrix, this entry is
20. So for these two matrices to be
equal, these two entries must be equal. In other words, by equating the
entries in row one and column one of both of our matrices, we get that 10𝑥 squared
plus 10 is equal to 20. We can then solve this equation for
𝑥. We’ll start by subtracting 10 from
both sides of the equation. This gives us that 10𝑥 squared is
equal to 10. Next, we’ll divide both sides of
the equation through by 10. This gives us that 𝑥 squared is
equal to one.
Finally, one way of solving this
equation is to take the square root of both sides. Remember, we’ll get a positive and
a negative square root. This gives us that 𝑥 is equal to
positive or negative one. So it doesn’t matter if 𝑥 is equal
to positive or negative one. Then the entries in row one and
column one of our matrices will be equal. However, we can’t stop there. We need to check whether 𝑥 appears
in the rest of the entries of our matrices because one of these solutions might not
be valid if it does. If we quickly check the rest of the
entries of our matrices, we can see none of them contain the variable 𝑥, so their
values are independent of 𝑥. So it doesn’t matter if 𝑥 is equal
to one or negative one when we’re checking the equality of these two matrices. The entries will be the same. So in actual fact, 𝑥 can be equal
to positive or negative one in this case.
Let’s now check the rest of the
entries in our two matrices. We can see in row one column two,
both of the entries are equal to two. In fact, we get the same story in
row two and column two. Both entries here are equal to
nine. In both of these cases, our
variables 𝑥 and 𝑦 don’t appear, so these will be equal regardless what we set
these values to.
The last entries we need to check
is row two column one. Again, remember, since we’re told
these two matrices are equal, their entries in row two column one must also be
equal. So by equating these two entries,
we get that negative three must be equal to two 𝑦 plus nine. And we can then solve this equation
for 𝑦. We’ll start by subtracting nine
from both sides of the equation. This gives us that negative 12 is
equal to two 𝑦. Now, what we need to do is divide
both sides of the equation through by two. We see that this gives us that 𝑦
is equal to negative six. Therefore, given that the matrix
10𝑥 squared plus 10, two, negative three, nine is equal to the matrix 20, two, two
𝑦 plus nine, nine, we were able to show that the value of 𝑥 must be equal to
positive or negative one and the value of 𝑦 must be equal to negative six.
