Question

In exercise 11 and 12, mark each statement True or False.Justify each answer.

a. If \(A = \left[ {\begin{array}{*{20}{c}}{{A_{\bf{1}}}}&{{A_{\bf{2}}}}\end{array}} \right]\) and \(B = \left[ {\begin{array}{*{20}{c}}{{B_{\bf{1}}}}&{{B_{\bf{2}}}}\end{array}} \right]\), with \({A_{\bf{1}}}\) and \({A_{\bf{2}}}\) the same sizes as \({B_{\bf{1}}}\) and \({B_{\bf{2}}}\), respectively then \(A + B = \left[ {\begin{array}{*{20}{c}}{{A_1} + {B_1}}&{{A_{\bf{2}}} + {B_{\bf{2}}}}\end{array}} \right]\).

b. If \(A = \left[ {\begin{array}{*{20}{c}}{{A_{{\bf{11}}}}}&{{A_{{\bf{12}}}}}\\{{A_{{\bf{21}}}}}&{{A_{{\bf{22}}}}}\end{array}} \right]\) and \(B = \left[ {\begin{array}{*{20}{c}}{{B_1}}\\{{B_{\bf{2}}}}\end{array}} \right]\), then the partitions of \(A\) and \(B\) are comfortable for block multiplication.

Short answer

a. True

b. False

Step-by-step solution

Substitute the expression \(A + B\)

Calculate the value of the expression \(A + B\).

\(A + B = \left[ {\begin{array}{*{20}{c}}{{A_1}}&{{A_2}}\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{{B_1}}&{{B_2}}\end{array}} \right]\)

Simplify the expression of \(A + B\)

Using the addition of matrices for \(A + B\), you get

\(\begin{array}{c}A + B = \left[ {\begin{array}{*{20}{c}}{{A_1}}&{{A_2}}\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}{{B_1}}&{{B_2}}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{{A_1} + {B_1}}&{{A_2} + {B_2}}\end{array}} \right].\end{array}\)

So, the given statement is true.

Check the block multiplication for \(A + B\)

Partition matrices can be multiplied by the usualrow-column rule. If the block entries are scalars when \(AB\) exists, the column partition of \(A\) matches the row partition of \(B\).

For the given matrices, the column partitionof \(A\) does not match with the row partition. Therefore, the given statement is false.

So, statement (a) is True, (b) is False.