Question

Suppose the second column of Bis all zeros. What can you

say about the second column of AB?

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 second column of Bis all zeros. So,\({{\bf{b}}_2} = {\bf{0}}\), and the matrix Bcan be represented as follows:

\(B = \left( {\begin{aligned}{*{20}{c}}{{{\bf{b}}_1}}&{\bf{0}}&{{{\bf{b}}_3}}& \cdots &{{{\bf{b}}_p}}\end{aligned}} \right)\)

Obtain the product AB as shown below:

\(\begin{aligned}{c}AB = A\left( {\begin{aligned}{*{20}{c}}{{{\bf{b}}_1}}&{\bf{0}}&{{{\bf{b}}_3}}& \cdots &{{{\bf{b}}_p}}\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}{A{{\bf{b}}_1}}&{A\left( {\bf{0}} \right)}&{A{{\bf{b}}_3}}& \cdots &{A{{\bf{b}}_p}}\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}{A{{\bf{b}}_1}}&{\bf{0}}&{A{{\bf{b}}_3}}& \cdots &{A{{\bf{b}}_p}}\end{aligned}} \right)\end{aligned}\)

So, if\({{\bf{b}}_2} = {\bf{0}}\), then\(A{{\bf{b}}_2} = {\bf{0}}\).

Thus, ifthe second column of Bis all zeros, thesecond column of the product AB is also zero.