Let A and B be arbitrary matrices for which the indicated product is defined. Determine whether the statement below is true or false. Justify the answer. If A and B are 2 x 2 with column a1,a2 and b1, b2 respectivley then AB= [a1b1, a2b2] Each column of AB is a linear combination of the columns of B using weights from the corresponding column of A.

a. True
b. False

b. False

Step-by-step explanation:

Let A and B be matrices with the rows a1a2 and b1b2 respectively. Suppose the other two columns are c1,c2 and d1d2 having a value equal to zero.

A=   a1   c1                      B=  b1       d1

      a2  c2                            b2      d2

Multiplying

AXB= a1b1+c1b2         a1d1+c1d2

          a2b1+c2b2       a2d1+c2d2

AB=║ a1b1         0║

      ║a2b1        0 ║

Hence the given statement is false.

Each column of AB is a linear combination of the ROWS of A using weights from the corresponding column of B.

The matrix multiplication is "row by column" i.e the rows of the first matrix are multiplied with the columns of the second matrix.

The rows are also used in multiplication.