WEBVTT
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In this video, we’ll learn what it means to say that two matrices are equal.
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We’ll identify some conditions which must be satisfied for two matrices to be equal.
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And we’ll use these conditions to solve equations based on equal matrices.
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Before we talk about matrices, let’s start by talking about equality because we’ve seen a lot of different types of equality before.
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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.
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So with numbers, it’s very easy to check if they’re equal.
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We just need to check if they’re the same number.
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But this is not the only type of equality we’ve seen.
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For example, consider the vector 𝐯 equal to two 𝐢 plus three 𝐣 and the vector 𝐮 equal to two 𝐢 plus three 𝐣.
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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.
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In this case, both coefficients of 𝐢 are equal to two and both coefficients of 𝐣 are equal to three.
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So 𝐯 is equal to 𝐮.
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But this involves more checks the moment we’re just using numbers.
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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.
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But then we know there’s also one more problem on top of this with vectors.
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Imagine if instead we had had the vector 𝐰 equal to two 𝐢 plus three 𝐣 plus 𝐤.
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Now we can see the coefficient of 𝐢 is equal to two and the coefficient of 𝐣 is equal to three.
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But we know our vector 𝐰 is still not equal to our vector 𝐯.
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This is because we have a third unit directional vector in 𝐰.
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𝐯 is a two-dimensional vector and 𝐰 is a three-dimensional vector, so they can’t be equal.
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So when we’re talking about equality, it won’t always be as simple as just checking whether two numbers are equal.
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We want to talk about the equality of two matrices.
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Remember, matrices, just like vectors, have multiple entries.
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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.
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Let’s now move on to our definition of the equality of two matrices.
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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.
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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.
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In other words, for two matrices to be equal, all of their entries must be identical.
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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.
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So just like with vectors, all we need to do is check whether all of our entries are identical.
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Let’s now move on to some examples.
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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?
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Let’s start by recalling what we mean when we say that two matrices are equal.
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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.
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Otherwise, we say that these matrices are not equal.
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So to check the two matrices are equal, we need to check that all of their entries are identical.
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Let’s start with matrix 𝐴.
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We can see this has two rows and three columns.
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So in our case, what would our values of 𝑎 𝑖𝑗 be?
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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.
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We can then do something similar for our matrix 𝐵.
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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.
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But now we can start to see our problem from our definition.
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The entries must be equal for all possible rows and columns.
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Our matrix 𝐴 has three columns.
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However, our matrix 𝐵 only has two columns.
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So these matrices can’t possibly be equal.
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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.
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So in this case, the matrix 𝐴 is not equal to the matrix 𝐵.
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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.
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In other words, if they have a different number of rows or columns, they can’t be equal.
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This means what we’ve shown is for two matrices to be equal, they must have the same number of rows or columns.
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In other words, they must have the same order.
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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.
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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?
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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.
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We can see that our matrix 𝐴 has two rows and two columns and the matrix 𝐵 also has two rows and two columns.
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This means to check whether 𝐴 is equal to 𝐵, all we need to do is check whether their entries are identical.
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Another way of saying this is we’ve shown that the matrix 𝐴 and the matrix 𝐵 have the same order.
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So to check that these two matrices are equal, we now need to check that all of their entries are identical.
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Remember, we only compare entries in the same position in each matrix.
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And if any of these are not equal, then we know that our matrices are not equal.
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Let’s start with the entry in row one and column one for both of our matrices.
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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.
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So these entries are identical.
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Remember, we need to check this for all of our entries.
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Let’s now move on to row two and column one.
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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.
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So again, these are both equal.
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But what happens when we move on to row one and column two for both of our matrices?
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In matrix 𝐴, this value is equal to three.
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However, in matrix 𝐵, this value is equal to negative three.
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So the entries in row one and column two are not equal.
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And remember for two matrices to be equal, we must have all of their entries are identical.
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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.
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So far, we’ve only seen matrices with at most three rows or three columns.
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However, the same principle extends to larger matrices.
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We just need to check all of our entries are identical and that the two matrices have the same order.
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We can do more with equality of matrices than just check whether two matrices are equal, as we’ll see in our next example.
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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.
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The question gives us two matrices which we are told are equal.
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We need to use this information to find the values of 𝑥 and 𝑦.
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Remember, for two matrices to be equal, entries in the same row and column must be identical.
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So, for example, we must have both entries in the first row and first column equal.
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In this case, they’re both equal to negative four.
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But this doesn’t really help us find the values of 𝑥 or 𝑦.
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However, what happens if we look at the values in row two column one?
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Remember, these must be equal.
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In our first matrix, the value in row two column one is 𝑥.
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And in our second matrix, the value in row two column one is eight.
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So for our matrices to be equal, these two entries must be equal.
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In other words, we must have 𝑥 is equal to eight.
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We’ll want to do something similar to help us find the value of 𝑦.
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We can see the only place 𝑦 appears is in our second matrix in row two column two.
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And for these two matrices to be equal, they must have the same value in row two column two.
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So we can just equate the entries in row two column two for both of these matrices.
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In other words, we must have negative seven is equal to 𝑦 minus six.
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And we can then just solve this equation for 𝑦.
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We’ll add six to both sides of the equation.
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And we see that this gives us that 𝑦 is equal to negative one.
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One thing that’s often worth doing in situations like this is substituting our value of 𝑦 back into our matrix.
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Remember, when we do this, we should get the entry of negative seven.
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So we’ll substitute 𝑦 is negative one into the expression in row two column two in our second matrix.
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This gives us negative one minus six.
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And we can evaluate this, and we get negative seven just as we expected.
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This just helps us check that our answer was correct.
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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.
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So now we’ve seen that given the equality of two matrices, we can find out information about unknown entries of these matrices.
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Let’s now look at another example of this.
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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.
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We need to find the values of 𝑥 and 𝑦 that makes these two matrices equal.
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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.
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We can see that for the matrices given to us in the question, both of them have two rows and two columns.
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So this doesn’t help us find the values of 𝑥 or 𝑦.
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Let’s instead use the fact that all entries in the same row and column must be equal.
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Let’s start with the first row and the first column.
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In our first matrix, this entry is 10𝑥 squared plus 10.
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In our second matrix, this entry is 20.
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So for these two matrices to be equal, these two entries must be equal.
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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.
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We can then solve this equation for 𝑥.
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We’ll start by subtracting 10 from both sides of the equation.
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This gives us that 10𝑥 squared is equal to 10.
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Next, we’ll divide both sides of the equation through by 10.
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This gives us that 𝑥 squared is equal to one.
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Finally, one way of solving this equation is to take the square root of both sides.
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Remember, we’ll get a positive and a negative square root.
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This gives us that 𝑥 is equal to positive or negative one.
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So it doesn’t matter if 𝑥 is equal to positive or negative one.
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Then the entries in row one and column one of our matrices will be equal.
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However, we can’t stop there.
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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.
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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.
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So it doesn’t matter if 𝑥 is equal to one or negative one when we’re checking the equality of these two matrices.
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The entries will be the same.
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So in actual fact, 𝑥 can be equal to positive or negative one in this case.
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Let’s now check the rest of the entries in our two matrices.
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We can see in row one column two, both of the entries are equal to two.
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In fact, we get the same story in row two and column two.
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Both entries here are equal to nine.
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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.
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The last entries we need to check is row two column one.
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Again, remember, since we’re told these two matrices are equal, their entries in row two column one must also be equal.
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So by equating these two entries, we get that negative three must be equal to two 𝑦 plus nine.
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And we can then solve this equation for 𝑦.
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We’ll start by subtracting nine from both sides of the equation.
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This gives us that negative 12 is equal to two 𝑦.
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Now, what we need to do is divide both sides of the equation through by two.
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We see that this gives us that 𝑦 is equal to negative six.
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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.
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Let’s now go through one more example on how we can use the equality of matrices to find the value of certain variables.
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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.
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Given that the matrix 𝐴 is equal to the matrix 𝐵, determine the values of 𝑥, 𝑦, and 𝑧.
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In this question, we’re given two matrices 𝐴 and 𝐵.
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And we can see that the entries in matrix 𝐴 are dependent on the three variables 𝑥, 𝑦, and 𝑧.
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In fact, we’re told that these two matrices are equal.
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We need to use this information to determine the values of 𝑥, 𝑦, and 𝑧.
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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.
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In fact, this also tells us that our matrices must have the same number of rows and columns.
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In our case, matrix 𝐴 is negative 10𝑥, 𝑥 plus three 𝑦, two 𝑥 minus 𝑧 and matrix 𝐵 is negative 30, 27, 10.
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Since we’re told that these two matrices are equal, entries in the same row and same column must be equal.
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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.
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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.
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And we can solve this equation for 𝑥.
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We just divide both sides by negative 10.
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And this gives us that our value of 𝑥 must be equal to three.
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Let’s now move on to row one and column two of our two matrices.
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In matrix 𝐴, we can see that the entry in row one column two is equal to 𝑥 plus three 𝑦.
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And in matrix 𝐵, the entry in row one column two is equal to 27.
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So because these two matrices are equal, these two entries must be equal.
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This gives us that 𝑥 plus three 𝑦 must be equal to 27.
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Remember, we already showed that our value of 𝑥 must be equal to three, so we can substitute this into our equation.
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This gives us that three plus three 𝑦 must be equal to 27.
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We can then solve this equation for 𝑦.
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We’ll start by subtracting three from both sides of the equation.
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This gives us that three 𝑦 is equal to 24.
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Now to solve this equation for 𝑦, we’ll divide both sides of the equation through by three.
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And so we get that 𝑦 is equal to 24 divided by three, which is, of course, equal to eight.
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We can do the same with the final entry in each of our two matrices, the entry in row one column three.
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In matrix 𝐴, this entry is two 𝑥 minus 𝑧.
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And in matrix 𝐵, this entry is 10.
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And remember, we’re told matrix 𝐴 is equal to matrix 𝐵, so we must have these two entries equal.
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So we get two 𝑥 minus 𝑧 is equal to 10.
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Remember, we already showed earlier that if our two matrices are equal, 𝑥 must be equal to three.
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So to help us find our value of 𝑧, we’ll substitute 𝑥 is equal to three into this equation.
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Substituting 𝑥 is equal to three, we get two times three minus 𝑧 is equal to 10.
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And we can just solve this equation for 𝑧.
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We add 𝑧 to both sides and then subtract 10 from both sides of the equation.
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This gives us that 𝑧 is equal to negative four.
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So this gives us that 𝑥 is equal to three, 𝑦 is equal to eight, and 𝑧 is equal to negative four.
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But remember, it can be very useful to substitute these values back into our matrix to check that our answer is correct.
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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.
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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.
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And if we evaluate each of these entries, we see we get negative 30, 27, 10.
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And each of these entries is exactly the same as we have in matrix 𝐵, so we know we have the right answer.
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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.
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So let’s now go over the key points of this video.
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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.
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Another way of saying this is all of the entries of our matrices must be identical.
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And this definition gave us some interesting properties.
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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.
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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.
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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.
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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.