Video: The Properties of Multiplication of Matrices

For 𝐴 = [4, −5 and 4, −5], write 𝐴² as a multiple of 𝐴.

03:56

Video Transcript

For 𝐴 equals the matrix four, negative five, four, negative five, write 𝐴 squared as a multiple of 𝐴.

Well, we know that 𝐴 squared is 𝐴 multiplied by 𝐴. So in that case, what we’re gonna do is multiply our matrix four, negative five, four, negative five by the same matrix. So as I said, we’re now gonna multiply our matrices together. And if we multiply a two-by-two matrix by a two-by-two matrix, we know that the result is gonna be also a two-by-two matrix.

So now, to find the first term in our matrix, so an answer matrix, what we’re gonna do is multiply the terms from the first row of the first matrix by the terms in the first column of the second matrix. And then we add them together. And what we’re actually doing is multiplying the corresponding terms. So, for instance, we’ve got the first term in the first row multiplied by the first term in the first column of the second matrix. And the second term from the first row of the first matrix by the second term in the first column of the second matrix. And then for the top-right term or the second term on the first row, we’re gonna do the same as before, but this time using the second column in our second matrix.

So when we do that, we get four multiplied by negative five plus negative five multiplied by negative five. And then we repeat the same pattern for the second row. So first of all, we have four multiplied by four plus negative five multiplied by four. And then as before, we move on to the second column for the bottom-right term. So we have four multiplied by negative five plus negative five multiplied by negative five.

So now, what we’re going to do is work these out. And when we do that, we get 16 plus negative 20. That’s cause we had four by four which is 16 plus negative five multiplied by four which gives us negative 20. Then we have negative 20 plus 25, 16 plus negative 20, and negative 20 plus 25 which leaves us with an answer matrix for 𝐴 squared of negative four, five, negative four, five.

So great, have we finished the question? Well, no because the question wants us to write 𝐴 squared as a multiple of 𝐴. Well, to do this, what we’re gonna do is compare our matrix 𝐴 with our matrix 𝐴 squared. So we have four, negative five, four, negative five and negative four, five, negative four, five. When we look at the terms, we can see that every term is the opposite sign of the equivalent term in the other matrix. So instead of four, we have negative four. Instead of negative five, we have positive five.

So, therefore, we can say that we’d multiply the matrix 𝐴 by negative one to reach the matrix 𝐴 squared. So let’s have a go doing that just to show that it’s correct. So we’ve got negative one multiplied by the matrix four, negative five, four, negative five. And if we do that, what it means is we multiply each of the terms by negative one. And when we do that, we get the matrix negative four, five, negative four, five. And that’s because four multiplied by negative one is negative four. And negative one multiplied by negative five — well, if you multiply two negatives, you get a positive — so it gives us positive five.

Well, this is the same as a matrix that we got for 𝐴 squared. So, therefore, we consider 𝐴 squared is equal to negative one multiplied by 𝐴. So, therefore, if the matrix 𝐴 is equal to four, negative five, four, negative five, then 𝐴 squared as a multiple of 𝐴 is gonna be equal to negative 𝐴.

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