Lesson Video: Equal Matrices Mathematics

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.