Lete_{j}denote the jth unit column that contains a 1 in the jth

position and zeros everywhere else. For a general matrix A_{n×n}, describe the following products. (a) Ae_{j }(c)e^{T}_{i}Ae_{j}?

Theorem:

Rows and Columns of a Product

Suppose thatA = [a_{ij}] is m × p andB = [b_{ij}] is p × n.

• [AB]_{i∗}=A_{i∗}B [( ith row ofAB)=( ith row ofA) ×B]. (3.5.4)

• [AB]_{∗j}=AB_{∗j}[ (jth col ofAB)=A× ( jth col ofB)]. (3.5.5)

• [AB]_{i∗}=a_{i1}B_{1∗}+a_{i2}B_{2∗}+ · · · +a_{ip}B_{p∗}=Ʃa_{ik}B_{k∗}. (3.5.6)

• [AB]_{∗j }=A_{∗1}b_{1j}+A_{∗2}b_{2j}+ · · · +A_{∗p}b_{pj}=ƩA_{∗k}b_{kj}(3.5.7)

These last two equations show that rows ofAB are combinations of

rows ofB, while columns ofAB are combinations of columns ofA.

For parts a and c im not even sure what they are even asking for. When its sayinge_{j}is a unit column does that mean like this (1 0 0...0) as an example?