Question Video: Finding the Square of a Given Matrix

Given that 𝐴 = the matrix −6, 1, −5, 5, find 𝐴².


Video Transcript

Given that 𝐴 equals the matrix negative six, one, negative five, five, find 𝐴 squared.

Okay, so to find 𝐴 squared, what we’re gonna be doing, is we’re gonna be multiplying the matrix negative six, one, negative five, five by itself. And I’ve set it out here so that we can actually see that I’m multiplying one matrix by another matrix. I’ve also circled here the dot between them. And that’s just to tell you that this dot, that actually means multiply, is telling us that we’re actually multiplying these two matrices together.

Now the first thing to know, is that when we multiply a two-by-two matrix by a two-by-two matrix, the answer is also gonna be a two-by-two matrix. Okay, now when we’re actually looking what we’re going to multiply, we’re gonna be multiplying rows by columns. So actually, what we’re gonna be doing is when we multiply this row by this column, is we’re actually gonna be multiplying the corresponding members. So for example, we’re gonna start with negative six multiplied by negative six cause that’s the first term in our row multiplied by the first term in our column. And then we’re going to add to it. And you always add. It’s gonna be one multiplied by negative five cause that’s the second term in our first row and our second term in our first column.

Okay, great. So we can now find out the value. So this is all equal to 31. And so that means, we’ve now found the first term in our matrix. Now we can move on to the second term. And this time what we’re gonna do is, we’re gonna multiply our first row by our second column. So we’ve got negative six, one multiplied by one, five. Again, we’re multiplying corresponding members. So for instance, we multiply the negative six by the one cause it’s the first term in our first row. And then this one is the first term in our second column. And then we’ve got one times five. And again, we add them. And we get the answer negative one. So this is gonna be the next term in our matrix.

Okay, so now for the bottom-left term, what we’re gonna do, is we’re gonna multiply our second row by our first column. So we’re gonna get negative five, five multiplied by negative six, negative five. Again, just cause I’m reminding, the-the dot we’re using here is the same. It means multiply. So it’s also called the dot product. So we’re gonna get negative five multiplied by negative six plus five multiplied by negative five, which gives us an answer of five. So great, one more term added to our matrix. Right for the final value, we’re gonna multiply the second row by the second column. So it’s gonna be negative five, five multiplied by one, five which is gonna give us negative five multiplied by one plus five multiplied by five, which is all equal to 20.

So therefore, we can say our final value is gonna be 20. And therefore, we can say our final answer is that 𝐴 squared is equal to the matrix 31, negative one, five, 20.

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