Video Transcript
Suppose the matrix ๐ด is equal to one, one, negative one, zero, negative two, one; the matrix ๐ต is negative two, one, negative three, zero; and the matrix ๐ถ is zero, negative two, three, one, one, zero. Which of the following products is defined? Weโve got (A) which is ๐ต๐ถ, (B) is ๐ถ squared, (C) ๐ด squared, (D) ๐ด๐ต, and (E) ๐ต๐ด.
So the first thing we need to consider when solving this problem is the dimensions or order of our matrices. So ๐ด is a two-by-three, ๐ต is a two-by-two, and ๐ถ is a three-by-two matrix. Well, you might think, well, how is this going to be useful? But letโs remind ourselves about multiplying matrices. Well, letโs imagine weโre multiplying two matrices with the dimensions ๐ by ๐ and ๐ by ๐.
Well, first of all, to allow them to actually be multiplied together, we need to have the second dimension of the first matrix the same as the first dimension of the second matrix. So we can see in our example here they are because theyโre both ๐. And then if they are, in fact, the same, so we can multiply our matrices together, then the dimensions of our result are gonna be the other two dimensions that we havenโt looked at. So in this case, itโs gonna be ๐ by ๐. Okay, great. So this is gonna help us solve our problem. And weโre gonna see how now because what weโre gonna do is take a look at the first possible answer (A), so ๐ต๐ถ.
So this is multiplying matrix ๐ต by matrix ๐ถ, so a two-by-two matrix by a three-by-two matrix. Well, we can see that the second dimension of the first matrix is not the same as the first dimension of the second matrix because they are two and three, respectively. So that means that ๐ต๐ถ is not defined. Well, then, if we take a look at possible answer (B), this is ๐ถ squared. So itโs matrix ๐ถ multiplied by matrix ๐ถ, so a three by two multiplied by a three by two. Well, we can see here, again, the two middle dimensions are not the same cause we have a two and a three. So this is not defined.
So now if we look at answer (C), weโve got ๐ด squared, which is ๐ด multiplied by ๐ด. So weโre gonna have a two-by-three matrix multiplied by a two-by-three matrix. So once again, the second dimension in the first matrix is not equal to the first dimension in the second matrix. So this would be undefined. Then once again, weโve got an undefined result with (D) because this is matrix ๐ด multiplied by a matrix ๐ต, so a two-by-three multiplied by a two-by-two matrix. So therefore, we can assume that answer (E) will be the correct answer, ๐ต๐ด. But letโs double-check just to make sure.
So then to check this, weโve got answer (E), ๐ต๐ด, which is the matrix ๐ต multiplied by the matrix ๐ด, so a two-by-two matrix multiplied by a two-by-three matrix. Well, here, we can see in fact yes, the second dimension of the first matrix is equal to the first dimension of the second matrix cause theyโre both two. So therefore, we can say that the product ๐ต๐ด is going to be defined. And in fact, what we can also deduce is that the result is gonna be a two-by-three matrix.