Exercise 1.2.3 (Practice with subscripts 2)

Instead of that one vector $a$, suppose you have $n$ vectors $a_1$ to $a_n$ in the columns of $A$. Suppose you have $n$ vectors $b_1^T,\dots ,b_n^T$ in the rows of $B$.

(a)

Give a "sum of rank one" formula for the matrix-matrix product $\mathit{AB}$.

(b)

Give a formula for the $i,j$ entry of that matrix-matrix product $\mathit{AB}$. Use sigma notation to add the $i,j$ entries of each matrix $a_k{b}_k^T$, found in Problem 2.