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# Using identity & zero matrices

## Video transcript

so we have a matrix equation set up right over here we have matrix a times this plus matrix B times this plus matrix C times this is equal to this 2 by 2 matrix and the way it's set up it's a little bit of a puzzle and I'll give you one clue as to as to what type of matrices a B and C could be there each either going to be an identity matrix identity matrix or there or a zero matrix so given that clue that each of these are either an identity matrix or a zero matrix can you pause this video and essentially solve this puzzle which of these are identity matrices and which of these are 0 matrices assuming you've given a go at it so let's let's go entry by entry so if you look at this this first entry right over here how can we get a 2 for this this top left entry well let's see when we multiply if if these are idaite if any of these are identities and essentially you just get the value of the matrix if any of these are 0 then essentially that product doesn't get added that where it's so it's one way of thinking about is that that that matrix won't matter anymore because going to be multiplied by 0 so if a was a 0 matrix and B and C were identity matrices you would add 1 plus 1 to get to see just like maybe that's the case but it could be the other way around it could be that a is identity matrix B is a 0 matrix and C is an identity matrix and you add 1 plus 1 over there to get 2 or you could say that maybe C is the 0 matrix and B is the identity matrix and you add 1 plus 1 here so really all this is is telling us is two of these matrix two of two of a B and C are going to be identity matrices and one is going to be a zero matrix but we don't know which one is which just yet any of them can be the zero matrix at least based on looking at that first entry now let's look at I don't know let's look at this entry right over here how can we add up to four well let's see 3 3 if this was an identity matrix and this was an identity matrix then you're going to then essentially you'll just be left with this matrix plus this matrix this is going to be the zero so three plus negative five that is not equal to four so both of the you can't have a and B B identity matrices and C be the zero matrix so let's think about other other combinations here so what about B and C being the identity identity matrices and a being the zero matrix in that situation so a is the zero matrix so that's not going to matter and we're essentially when you multiply B times this you're just going to get this matrix and C times this you're going to get that matrix if B and C are identity matrices and so you have negative 5 plus 1 would be negative 4 so that doesn't work either so our last scenario is going to be so we can essentially rule out B as an identity matrix because when we took when we when we had B be one of the identity matrices and we picked the other two options we still couldn't get to 4 here so B is going to be a zero matrix and let's see if that works out a is identity then a would be an identity and C would be an energy matrix as well so let's see if that is if that actually makes sense so if this is if this right over here is identity matrix and that over there is an identity matrix this whole thing simplifies to 1 3 4 comma negative 2 plus 1 1 3 2 and it does indeed equal 1 plus 1 is 2 3 plus 1 is 4 4 plus 3 is 7 negative 2 plus 2 is indeed 0