Problem #9: Consider the following matrices. A = [1 0 -3 3;...

Problem #9: Consider the following matrices. A = [1 0 -3 3; -1 3 1 3], B = [1 0 -1; 3 4 -1; 2 -4 1], C = [1 0; -2 -2; 2 -4] (a) Let D = B^T C. Find d_32. (i.e., find the entry in row 3, column 2 of the matrix D) (b) Find tr(2BB^T) (c) Let E = B(CA). Find e_14. (i.e., find the entry in row 1, column 4 of the matrix E)

Transcript

00:01 We consider the following matrices a equal 1 0 negative 3 3 first row and second row negative 1 3 1 3 b is square matrix 1 0 negative 1 3 4 negative 1 and 2 negative 4 1 and c is the matrix 1 0 negative 2 negative 2 and 2 negative 4 so in part a we let d be the matrix b transpose times c we want to find the entry d 3 2 that is the entry in row 3 column 2 of matrix d in part b we find the trace of 2 times b times b transpose in part c we let e be the matrix b times c times a we want to find e 1 4 that is the entry in row 1 column 4 of matrix e good so a key thing in this exercise is not to do all the calculations that is you don't have to calculate the entire matrix b transpose c in order to find d 3 2 of course you can do it but you will do a lot of work that is not necessary same thing for the trace of 2 times b times b transpose and here for this matrix of course the part c will have more work than parts a and b but it won't be the work the complete work of calculating c times a and then b times that result c times a.

01:42 Okay so let's see part a to explain what we're talking about here so we got to remember the definition of the an entry in the product matrix that is d is the product of two matrices b transpose and c and so when we talk about element d 3 2 we obtain this element by doing the inner product of the third row of the first factor b transpose and the second column of the second factor matrix c and i'm going to write down that here in order to know that that is d 3 2 is the dot product between the third row of b transpose and the second column of c and that's always true that is when you have product of two matrices any element in that product matrix is obtained by doing an inner product the first vector is given by the first index of the entry we want to calculate and that first entry corresponds to the row of the first factor and the second index of the element we are looking for corresponds to the column of the second factor in the product of matrices so that's always like that.

03:37 Okay so you have that you know this but you know more now you can even say that d 3 2 is then you're talking about here the third row of b transpose but remember that b transpose is obtained from b by changing rows into columns so the third row of b transpose will be the second the third row the third column of b...

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