Video: Adding and Subtracting the Multiplication of Two Given Matrices

Consider 𝑋 = [−3, −3 and 5, −6], 𝑌 = [1, 3 and 6, −6]. What is 𝑋² − 𝑌²?

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Video Transcript

Consider the matrix 𝑋 equals negative three, negative three, five, negative six and 𝑌 equals one, three, six, negative six. What is 𝑋 squared minus 𝑌 squared?

Well, in order to solve this problem, what we’re gonna look at first of all is what 𝑋 squared and 𝑌 squared are separately. Well, first of all, if we start with 𝑋 squared, this is equal to the matrix negative three, negative three, five, negative six multiplied by the matrix negative three, negative three, five, negative six. And we know that we can multiply them together. Because when we look at the matrices, you can only multiply two matrices if and only if the number of columns in the first matrix equals the number of rows in the second matrix. And as they’re both two by two, this is definitely the case.

So in order to work out the first element in our matrix, what we’re going to do is we’re going to multiply the corresponding elements in the first row of the first matrix by the corresponding elements in the first column of the second matrix. So first of all, we’re gonna have negative three multiplied by negative three. And then what we’re gonna add to this is the product of the second element in the first row of the first matrix and the second element of the first column in the second matrix. So it’s gonna be negative three multiplied by five.

So then we’re gonna move on to the next element. And so we’re gonna go to the second element in the first row of our answer matrix. And then what we’ve got here is the first element of the first row of the first matrix multiplied by the first element of the second column of the second matrix. So negative three multiplied by negative three. And then we’re gonna add to this negative three multiplied by negative six. Okay, great. So that’s the first row of our answer matrix complete.

So then what we have is five multiplied by negative three. This is the bottom row of our answer matrix. Because what we have here is the first element in the second row of the first matrix multiplied by the first element in the first column of the second matrix. And then we add to this negative six multiplied by five. And then for the final element, we have five multiplied by negative three add negative six multiplied by negative six.

Okay, right. All we need to do now is calculate each of our elements. So our first element is gonna be negative six. That’s cause we had nine add negative 15. Well, if we add a negative, it’s the same as subtracting a positive. So this is gonna be negative six. And then if we move along, we’ve got nine add 18, which is 27. Then we’ve got negative 15 add negative 30, which is negative 45. And then, finally, we’ve got negative 15 add 36, which is 21. So therefore, 𝑋 squared is equal to the matrix negative six, 27, negative 45, and 21.

Okay, great. So now let’s move on to 𝑌 squared. Well, if we’re looking at 𝑌 squared, that’s gonna be equal to the matrix one, three, six, negative six multiplied by the matrix one, three, six, negative six. So using the same method as last time, our first element is gonna be one multiplied by one add three multiplied by six. Then we go to the next element, which will be one multiplied by three add three multiplied by negative six. And then for the second row, we’ve got six multiplied by one add negative six multiplied by six. Then for the final element, we’ve got six multiplied by three add negative six multiplied by negative six.

And then we calculate each element. So the first one is gonna be 19 because it’s one add 18. And then we’ve got three add negative 18, which is gonna give us negative 15. Then we’ve got six add negative 36, which is negative 30. And then, finally, we’ve got 18 add 36, which is 54. So therefore, we can say that the matrix 𝑌 squared is gonna be equal to 19, negative 15, negative 30, and 54.

Okay, great. So now, all we need to do is subtract this from 𝑋 squared. So when we do that, what we’re gonna have is the matrix negative six, 27, negative 45, 21 minus the matrix 19, negative 15, negative 30, 54. And to do this, what we do is we subtract the corresponding elements. So we’re gonna have negative six minus 19 and then 27 minus negative 15, then negative 45 minus negative 30, and then finally 21 minus 54. And this is gonna give us our final answer, which is a matrix negative 25, 42, negative 15, and negative 33.

And just one bit to be careful with, with our calculation here, is the bottom-left element. What we got is negative 45 minus negative 30. Remember, if you subtract a negative, it’s the same as adding a positive. So it’s negative 45 add 30, which gives us our negative 15.

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