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Current time:0:00Total duration:3:30

Using properties of matrix operations

Video transcript

in order to get into battle school cadets have to pass a rigorous entrance exam which includes mathematics help commander Graff grade the next wave of students tests the last step of a problem in the matrix multiplication section is the matrix a times B times C where a B and C are square matrices which of the following candidates have answers that are equivalent to this expression select all who are right for any a B and C so select all that apply so I encourage you to pause this video and think about it which of these expressions for any square matrices a B and C are equivalent to this right over here so I'm assuming you've given a go at it so let's think through each of them so this one is B AC so if they have changed the order and we've already seen that matrix multiplication is not commutative in general and so this will not be true for any square set of matrices a B and C so this is not going to be true matrix multiplication is not commutative so here you have Bernard who says a times C times B well we already know that that's going to be the equivalent to a C B which once again they've swapped the order between the B and the C matrix multiplication is not commutative you can't just swap order and expect to get the same product for any square matrices a B and C so we could rule that one out a times B C so we've already seen matrix multiplication is associative so this is the same thing as a times B times C which of course is the same thing as a b c so what Carn has right over here that is right that is equivalent for any square matrices a B and C that is equivalent to a b c now du cheville let's see now this looks like a bit of a crazy expression but let's think it through a little bit so first of all matrix multiplication as long as you keep the order right the distributive property does hold so this first part right over here is equivalent to two so let me write this down this one's interesting so we have we have a times b c plus a minus a squared so you can actually distribute this a and I encourage you to prove it for yourself maybe using some two-by-two matrices for for simplicity and so this is going to be equal to this part right here is going to be a b c plus a a a times a which we could write as a squared and then we're going to subtract a squared well these two things are going to cancel out they're going to end up being the zero matrix and if you take the zero so these are going to be the zero matrix right over here and if you take the zero matrix and add it to a b c you're just going to end up with you're just going to end up with a b c so this one was a little bit tricky this one actually is equivalent so this one so this one is right and this one is right and then here this is a times B plus C so this is kind of a kind of kind of kind of nutty right over here this is they're not even multiplying B and C so this one's definitely not going to be the true for all square matrices a B and C