Multiplying matrices

we're given two matrices over here matrix E and matrix D and they ask us what is e D which is another way of saying what is the product of matrix E and matrix D so just so I remember what I'm doing let me copy and paste this and then I'm going to get out my and then I'm going to get out my little scratch pad so let me paste that over here so we have all the information we needed and so let's try to work this out so matrix E times matrix D so this is matrix E times matrix matrix D which is equal to matrix E is all of this business so it is zero three five five five two times matrix D which is all of this we're going to multiply it times three three four four negative two negative two now the first thing that we have to check is whether this is even a valid operation now the matrix multiplication is a human defined operation that just happens in fact all operations are that happen to have neat properties now the way that us humans have defined matrix multiplication it only works if when we're multiplying our two matrices so this right over here has two rows and three columns so it's a two by three matrix and this has three rows and two columns it's three by two this only works we can only multiply this matrix times this matrix if the number of columns on this matrix is equal to the number of rows on this matrix and in this situation it is so I can actually multiply them if these two numbers weren't equal if the number of columns here were not equal to the number of rows here then this would not be a a valid operation at least the way that we have defined matrix multiplication the other thing you always have to remember is that eat iams D is not always the same thing as D times e order matters when you're multiplying matrices it doesn't matter if you're multiplying regular numbers but it matters for matrices but let's actually work this out so we're going to get what we're going to get is actually going to be a two by two matrix but I'm going to create some space here because we gonna have to do some computation so this is going to be equal to this is going to be equal to I'm going to make a huge 2x2 matrix here so the way we get the top left entry the top left entry is essentially going to be it's going to be this row it's going to be this row times this product if you viewed them each is vectors and you have some familiar with the dot product we're since you can take the dot product of that and that if you have no idea what that is I'm about to show you this entries going to be 0 times 3 so 0 times 3 plus 3 times 3 plus 3 times 3 plus 5 times 4 plus 5 times 4 so that is the top left entry and I already see that I'm going to run out of space here so let me move this over so let me move this over to the right some space so I have some breathing room now we can do the top right entry this was the top left now we're going to do the top right so the top right entry is going to be is going to be this row this row times this column this row times this column notice the entry is getting the row from the first matrix and the column from the second one that's kind of determining its position so once again it's going to be 0 times 4 0 times 4 plus 3 times negative 2 3 times negative 2 plus 5 times negative 2 5 times negative 2 and we keep going the bottom left entry is going to be this row the second row here times the first column here so it's 5 times 3 so 5 times 3 plus 5 times 3 plus 5 times 3 plus 2 times 4 plus 2 times 4 and we're almost done we just need to multiply or take the dot product of this row with this column right over here so it's going to be 5 times 4 5 times 4 plus 5 times negative 2 plus 2 times negative 2 2 times negative 2 and so this is going to be equal to and we can just evaluate this now let's see 0 times 3 is 0 this is 9 9 plus 20 is this is 29 this this all simplified to 29 all of this this is 0 this is negative 6 and then this is minus 10 so this all simplifies to negative 16 negative 16 this right over here it's 15 plus 15 which is 30 plus 8 so this is 38 38 and then finally this is 20 20 minus 10 minus 4 so this is going to be 6 so this is all going to simplify to 6 so this all became 29 negative 16 38 and 629 negative 16 thirty eight and six so 29 29 negative sixteen thirty eight and six let's check our answer and we got it right