Video Transcript
In this video, we’ll learn what it
means to say that two matrices are equal. We’ll identify some conditions
which must be satisfied for two matrices to be equal. And we’ll use these conditions to
solve equations based on equal matrices.
Before we talk about matrices,
let’s start by talking about equality because we’ve seen a lot of different types of
equality before. For example, we know if 𝑥 is equal
to five and 𝑦 is equal to five, then we must have that 𝑥 is equal to 𝑦 because
they both represent the same number, in this case, five. So with numbers, it’s very easy to
check if they’re equal. We just need to check if they’re
the same number.
But this is not the only type of
equality we’ve seen. For example, consider the vector 𝐯
equal to two 𝐢 plus three 𝐣 and the vector 𝐮 equal to two 𝐢 plus three 𝐣. In this case, to check that our
vectors are equal, we need to check that both the coefficients of 𝐢 are equal and
both the coefficients of 𝐣 are equal. In this case, both coefficients of
𝐢 are equal to two and both coefficients of 𝐣 are equal to three. So 𝐯 is equal to 𝐮. But this involves more checks the
moment we’re just using numbers. For example, if we have the vector
𝐰 is equal to two 𝐢 plus four 𝐣, then because the coefficient of 𝐣 in 𝐯 and 𝐰
are different, we must have that the vectors 𝐰 and 𝐯 are not equal.
But then we know there’s also one
more problem on top of this with vectors. Imagine if instead we had had the
vector 𝐰 equal to two 𝐢 plus three 𝐣 plus 𝐤. Now we can see the coefficient of
𝐢 is equal to two and the coefficient of 𝐣 is equal to three. But we know our vector 𝐰 is still
not equal to our vector 𝐯. This is because we have a third
unit directional vector in 𝐰. 𝐯 is a two-dimensional vector and
𝐰 is a three-dimensional vector, so they can’t be equal. So when we’re talking about
equality, it won’t always be as simple as just checking whether two numbers are
equal.
We want to talk about the equality
of two matrices. Remember, matrices, just like
vectors, have multiple entries. So we’ll have a lot of similarities
to defining the equality of two matrices as we did when defining the equality of two
vectors.
Let’s now move on to our definition
of the equality of two matrices. If we let 𝐴 and 𝐵 be matrices
which are described by the entries as follows, for matrix 𝐴, we’ll call the entry
in row 𝑖 and column 𝑗 𝑎 𝑖𝑗 and for matrix 𝐵, we’ll call the entry in row 𝑖
and column 𝑗 𝑏 𝑖𝑗. Then if we have 𝑎 𝑖𝑗 is equal to
𝑏 𝑖𝑗 for all of our values of 𝑖 and 𝑗, we say that our matrix 𝐴 is equal to
our matrix 𝐵. In other words, for two matrices to
be equal, all of their entries must be identical.
It’s worth noting if any of the
entries are not identical — for example, if the entries in row 𝑖 and column 𝑗 are
not equal, so there is an 𝑖 and a 𝑗 such that 𝑎 𝑖𝑗 is not equal to 𝑏 𝑖𝑗 —
then we say the matrix 𝐴 is not equal to the matrix 𝐵. So just like with vectors, all we
need to do is check whether all of our entries are identical. Let’s now move on to some
examples.
Given that 𝐴 is the matrix with
first row three, three, three and second row three, three, three and 𝐵 is the
matrix with first row three, three and second row three, three, is it true that the
matrix 𝐴 is equal to the matrix 𝐵?
Let’s start by recalling what we
mean when we say that two matrices are equal. If we have the entry in row 𝑖 and
column 𝑗 of matrix 𝐴 is 𝑎 𝑖𝑗 and the entry in row 𝑖 and column 𝑗 of matrix 𝐵
is 𝑏 𝑖𝑗, then if 𝑎 𝑖𝑗 is equal to 𝑏 𝑖𝑗 for all of our values of 𝑖 and 𝑗,
we say that the matrix 𝐴 is equal to the matrix 𝐵. Otherwise, we say that these
matrices are not equal. So to check the two matrices are
equal, we need to check that all of their entries are identical.
Let’s start with matrix 𝐴. We can see this has two rows and
three columns. So in our case, what would our
values of 𝑎 𝑖𝑗 be? First, our matrix 𝐴 has two rows
and three columns, so our values of 𝑖 range from one to two and our values of 𝑗
range from one to three. We can then do something similar
for our matrix 𝐵. If 𝑏 𝑘𝑙 is the entry in matrix
𝐵 in row 𝑘 and column 𝑙, then because our matrix 𝐵 only has two rows and two
columns, our values of 𝑘 range from one to two and our values of 𝑙 also range from
one to two. But now we can start to see our
problem from our definition. The entries must be equal for all
possible rows and columns. Our matrix 𝐴 has three
columns. However, our matrix 𝐵 only has two
columns. So these matrices can’t possibly be
equal.
For example, if we highlight the
two entries in matrix 𝐴 in column three, by our definition of equality, we would
have to have a third column in matrix 𝐵 which is equal to this column. So in this case, the matrix 𝐴 is
not equal to the matrix 𝐵.
In fact, we can use exactly the
same line of reasoning as we did in this question to deduce that if two matrices
have different orders, they can’t be equal. In other words, if they have a
different number of rows or columns, they can’t be equal. This means what we’ve shown is for
two matrices to be equal, they must have the same number of rows or columns. In other words, they must have the
same order.
Of course, just because two
matrices do have the same order does not mean they’re equal, as we’ll show in our
next example.
If 𝐴 is the matrix negative five,
three, negative seven, negative three and 𝐵 is the matrix negative five, negative
three, negative seven, three, is it true that the matrix 𝐴 is equal to the matrix
𝐵?
We recall for two matrices to be
equal, they need to have the same number of rows and columns and all of their
entries must be identical. We can see that our matrix 𝐴 has
two rows and two columns and the matrix 𝐵 also has two rows and two columns. This means to check whether 𝐴 is
equal to 𝐵, all we need to do is check whether their entries are identical. Another way of saying this is we’ve
shown that the matrix 𝐴 and the matrix 𝐵 have the same order. So to check that these two matrices
are equal, we now need to check that all of their entries are identical. Remember, we only compare entries
in the same position in each matrix. And if any of these are not equal,
then we know that our matrices are not equal.
Let’s start with the entry in row
one and column one for both of our matrices. We see the entry in row one and
column one of matrix 𝐴 is negative five and the entry in row one and column one of
matrix 𝐵 is also negative five. So these entries are identical. Remember, we need to check this for
all of our entries. Let’s now move on to row two and
column one. This time, we see the entry in row
two and column one of matrix 𝐴 is negative seven and the entry in row two and
column one of matrix 𝐵 is also negative seven. So again, these are both equal.
But what happens when we move on to
row one and column two for both of our matrices? In matrix 𝐴, this value is equal
to three. However, in matrix 𝐵, this value
is equal to negative three. So the entries in row one and
column two are not equal. And remember for two matrices to be
equal, we must have all of their entries are identical. Therefore, given 𝐴 is equal to
negative five, three, negative seven, negative three and 𝐵 is equal to negative
five, negative three, negative seven, three because they have differing entries in
row one column two, we were able to conclude the matrix 𝐴 is not equal to matrix
𝐵.
So far, we’ve only seen matrices
with at most three rows or three columns. However, the same principle extends
to larger matrices. We just need to check all of our
entries are identical and that the two matrices have the same order. We can do more with equality of
matrices than just check whether two matrices are equal, as we’ll see in our next
example.
Given that the matrix negative
four, three, 𝑥, negative seven is equal to the matrix negative four, three, eight,
𝑦 minus six, find the values of 𝑥 and 𝑦.
The question gives us two matrices
which we are told are equal. We need to use this information to
find the values of 𝑥 and 𝑦. Remember, for two matrices to be
equal, entries in the same row and column must be identical. So, for example, we must have both
entries in the first row and first column equal. In this case, they’re both equal to
negative four. But this doesn’t really help us
find the values of 𝑥 or 𝑦. However, what happens if we look at
the values in row two column one? Remember, these must be equal. In our first matrix, the value in
row two column one is 𝑥. And in our second matrix, the value
in row two column one is eight. So for our matrices to be equal,
these two entries must be equal. In other words, we must have 𝑥 is
equal to eight.
We’ll want to do something similar
to help us find the value of 𝑦. We can see the only place 𝑦
appears is in our second matrix in row two column two. And for these two matrices to be
equal, they must have the same value in row two column two. So we can just equate the entries
in row two column two for both of these matrices. In other words, we must have
negative seven is equal to 𝑦 minus six. And we can then just solve this
equation for 𝑦. We’ll add six to both sides of the
equation. And we see that this gives us that
𝑦 is equal to negative one.
One thing that’s often worth doing
in situations like this is substituting our value of 𝑦 back into our matrix. Remember, when we do this, we
should get the entry of negative seven. So we’ll substitute 𝑦 is negative
one into the expression in row two column two in our second matrix. This gives us negative one minus
six. And we can evaluate this, and we
get negative seven just as we expected. This just helps us check that our
answer was correct. Therefore, given the matrix
negative four, three, 𝑥, negative seven is equal to the matrix negative four,
three, eight, 𝑦 minus six, we were able to show that the value of 𝑥 must be eight
and the value of 𝑦 must be negative one.
So now we’ve seen that given the
equality of two matrices, we can find out information about unknown entries of these
matrices. Let’s now look at another example
of this.
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.
Let’s now go through one more
example on how we can use the equality of matrices to find the value of certain
variables.
Consider the matrix 𝐴 is equal to
negative 10𝑥, 𝑥 plus three 𝑦, two 𝑥 minus 𝑧 and the matrix 𝐵 is equal to
negative 30, 27, 10. Given that the matrix 𝐴 is equal
to the matrix 𝐵, determine the values of 𝑥, 𝑦, and 𝑧.
In this question, we’re given two
matrices 𝐴 and 𝐵. And we can see that the entries in
matrix 𝐴 are dependent on the three variables 𝑥, 𝑦, and 𝑧. In fact, we’re told that these two
matrices are equal. We need to use this information to
determine the values of 𝑥, 𝑦, and 𝑧. Remember, we say that a matrix 𝐴
is equal to a matrix 𝐵 if all of the entries in the same row and same column are
equal. In fact, this also tells us that
our matrices must have the same number of rows and columns.
In our case, matrix 𝐴 is negative
10𝑥, 𝑥 plus three 𝑦, two 𝑥 minus 𝑧 and matrix 𝐵 is negative 30, 27, 10. Since we’re told that these two
matrices are equal, entries in the same row and same column must be equal. For example, the entry in the first
row and first column of matrix 𝐴 is negative 10𝑥, and the entry in the first row
and first column of matrix 𝐵 is negative 30. So for these two matrices to be
equal, these two entries must be equal, so we get negative 10𝑥 must be equal to
negative 30. And we can solve this equation for
𝑥. We just divide both sides by
negative 10. And this gives us that our value of
𝑥 must be equal to three.
Let’s now move on to row one and
column two of our two matrices. In matrix 𝐴, we can see that the
entry in row one column two is equal to 𝑥 plus three 𝑦. And in matrix 𝐵, the entry in row
one column two is equal to 27. So because these two matrices are
equal, these two entries must be equal. This gives us that 𝑥 plus three 𝑦
must be equal to 27. Remember, we already showed that
our value of 𝑥 must be equal to three, so we can substitute this into our
equation. This gives us that three plus three
𝑦 must be equal to 27. We can then solve this equation for
𝑦. We’ll start by subtracting three
from both sides of the equation. This gives us that three 𝑦 is
equal to 24. Now to solve this equation for 𝑦,
we’ll divide both sides of the equation through by three. And so we get that 𝑦 is equal to
24 divided by three, which is, of course, equal to eight.
We can do the same with the final
entry in each of our two matrices, the entry in row one column three. In matrix 𝐴, this entry is two 𝑥
minus 𝑧. And in matrix 𝐵, this entry is
10. And remember, we’re told matrix 𝐴
is equal to matrix 𝐵, so we must have these two entries equal. So we get two 𝑥 minus 𝑧 is equal
to 10. Remember, we already showed earlier
that if our two matrices are equal, 𝑥 must be equal to three. So to help us find our value of 𝑧,
we’ll substitute 𝑥 is equal to three into this equation. Substituting 𝑥 is equal to three,
we get two times three minus 𝑧 is equal to 10. And we can just solve this equation
for 𝑧. We add 𝑧 to both sides and then
subtract 10 from both sides of the equation. This gives us that 𝑧 is equal to
negative four. So this gives us that 𝑥 is equal
to three, 𝑦 is equal to eight, and 𝑧 is equal to negative four.
But remember, it can be very useful
to substitute these values back into our matrix to check that our answer is
correct. So let’s substitute 𝑥 is equal to
three, 𝑦 is equal to eight, and 𝑧 is equal to negative four into our matrix
𝐴. Substituting these values in, we
get that our matrix 𝐴 is equal to negative 10 times three, three plus three times
eight, two times three minus negative four. And if we evaluate each of these
entries, we see we get negative 30, 27, 10. And each of these entries is
exactly the same as we have in matrix 𝐵, so we know we have the right answer.
Therefore, if the matrix 𝐴 is
equal to negative 10𝑥, 𝑥 plus three 𝑦, two 𝑥 minus 𝑧 and the matrix 𝐵 is equal
to negative 30, 27, 10, then for 𝐴 to be equal to 𝐵, we must have that 𝑥 is equal
to three, 𝑦 is equal to eight, and 𝑧 is equal to negative four.
So let’s now go over the key points
of this video. We showed if we have two matrices,
first the matrix 𝐴 where the entry in row 𝑖 and column 𝑗 is given by 𝑎 𝑖𝑗 and
second the matrix 𝐵, where the entry in row 𝑖 and column 𝑗 of 𝐵 is given by 𝑏
𝑖𝑗, then for these two matrices to be equal, we must have that 𝑎 𝑖𝑗 is equal to
𝑏 𝑖𝑗 for all of our values of 𝑖 and 𝑗. Another way of saying this is all
of the entries of our matrices must be identical. And this definition gave us some
interesting properties.
For example, if for some 𝑖 and
some 𝑗 we have that a 𝑖𝑗 is not equal to 𝑏 𝑖𝑗, then we can say the matrix 𝐴
is not equal to the matrix 𝐵. In other words, we only needed one
entry in row 𝑖 and column 𝑗 for both of our matrices to be unequal for our
matrices to not be equal. Another consequence of this
definition we showed is if our matrices 𝐴 and 𝐵 have different orders, then matrix
𝐴 can’t be equal to matrix 𝐵. And another way of saying this was
to say that if our matrices have a different number of rows or a different number of
columns, then they can’t be equal.
