Suppose the matrix 𝐴 is equal to one, one, negative one, zero, negative two, one; the matrix 𝐵 is negative two, one, negative three, zero; and the matrix 𝐶 is zero, negative two, three, one, one, zero. Which of the following products is defined? We’ve got (A) which is 𝐵𝐶, (B) is 𝐶 squared, (C) 𝐴 squared, (D) 𝐴𝐵, and (E) 𝐵𝐴.
So the first thing we need to consider when solving this problem is the dimensions or order of our matrices. So 𝐴 is a two-by-three, 𝐵 is a two-by-two, and 𝐶 is a three-by-two matrix. Well, you might think, well, how is this going to be useful? But let’s remind ourselves about multiplying matrices. Well, let’s imagine we’re multiplying two matrices with the dimensions 𝑚 by 𝑛 and 𝑛 by 𝑝.
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 they’re both 𝑛. 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 haven’t looked at. So in this case, it’s gonna be 𝑚 by 𝑝. Okay, great. So this is gonna help us solve our problem. And we’re gonna see how now because what we’re gonna do is take a look at the first possible answer (A), so 𝐵𝐶.
So this is multiplying matrix 𝐵 by matrix 𝐶, 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 𝐵𝐶 is not defined. Well, then, if we take a look at possible answer (B), this is 𝐶 squared. So it’s matrix 𝐶 multiplied by matrix 𝐶, 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), we’ve got 𝐴 squared, which is 𝐴 multiplied by 𝐴. So we’re 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, we’ve got an undefined result with (D) because this is matrix 𝐴 multiplied by a matrix 𝐵, 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, 𝐵𝐴. But let’s double-check just to make sure.
So then to check this, we’ve got answer (E), 𝐵𝐴, which is the matrix 𝐵 multiplied by the matrix 𝐴, 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 they’re both two. So therefore, we can say that the product 𝐵𝐴 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.