Question

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

Short answer

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.

Step-by-step solution

Definition of matrix multiplication

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:

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

The matrix product AB

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.