Suppose the third column of Bis the sum of the first two columns. What can you say about the third column of AB? Why?

Consider two matrices P and Q of order\(m \times n\)and\(n \times p\), respectively. The order of the product PQ matrix is\(m \times p\).

Let\({{\bf{q}}_1},{{\bf{q}}_2},...,{{\bf{q}}_n}\)be the columns of the matrix Q. Then, the product PQ is obtained as shown below:

\(\begin{aligned}{c}PQ = P\left( {\begin{aligned}{*{20}{c}}{{{\bf{q}}_1}}&{{{\bf{q}}_2}}& \cdots &{{{\bf{q}}_n}}\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}{P{{\bf{q}}_1}}&{P{{\bf{q}}_2}}& \cdots &{P{{\bf{q}}_n}}\end{aligned}} \right)\end{aligned}\)

It is given thatthe third column of Bis the sum of the first two columns. So,\({{\bf{b}}_3} = {{\bf{b}}_1} + {{\bf{b}}_2}\), and the matrix Bcan be represented as follows:

\(B = \left( {\begin{aligned}{*{20}{c}}{{{\bf{b}}_1}}&{{{\bf{b}}_1}}&{{{\bf{b}}_1} + {{\bf{b}}_2}}& \cdots &{{{\bf{b}}_p}}\end{aligned}} \right)\)

Obtain the product AB as shown below:

\(\begin{aligned}{c}AB = A\left( {\begin{aligned}{*{20}{c}}{{{\bf{b}}_1}}&{{{\bf{b}}_1}}&{{{\bf{b}}_1} + {{\bf{b}}_2}}& \cdots &{{{\bf{b}}_p}}\end{aligned}} \right)\\ = \left( {A\begin{aligned}{*{20}{c}}{{{\bf{b}}_1}}&{A{{\bf{b}}_1}}&{A\left( {{{\bf{b}}_1} + {{\bf{b}}_2}} \right)}& \cdots &{A{{\bf{b}}_p}}\end{aligned}} \right)\\ = \left( {A\begin{aligned}{*{20}{c}}{{{\bf{b}}_1}}&{A{{\bf{b}}_1}}&{A{{\bf{b}}_1} + A{{\bf{b}}_2}}& \cdots &{A{{\bf{b}}_p}}\end{aligned}} \right)\end{aligned}\)

It is observed that the third column of AB, that is,\(A{{\bf{b}}_3} = A{{\bf{b}}_1} + A{{\bf{b}}_2}\)is the sum of the first two columns of AB.

Thus, if thethird column of Bis the sum of the first two columns,the third column of AB is the sum of the first two columns of AB.