Question

In Exercises 1 and 2, compute each matrix sum or product if it is defined. If an expression is undefined, explain why. Let

\(A = \left( {\begin{aligned}{*{20}{c}}2&0&{ - 1}\\4&{ - 5}&2\end{aligned}} \right)\), \(B = \left( {\begin{aligned}{*{20}{c}}7&{ - 5}&1\\1&{ - 4}&{ - 3}\end{aligned}} \right)\), \(C = \left( {\begin{aligned}{*{20}{c}}1&2\\{ - 2}&1\end{aligned}} \right)\), \(D = \left( {\begin{aligned}{*{20}{c}}3&5\\{ - 1}&4\end{aligned}} \right)\) and \(E = \left( {\begin{aligned}{*{20}{c}}{ - 5}\\3\end{aligned}} \right)\)

\(A + 2B\), \(3C - E\), \(CB\), \(EB\).

Short answer

\(A + 2B = \left( {\begin{aligned}{*{20}{c}}{16}&{ - 10}&1\\6&{ - 13}&{ - 4}\end{aligned}} \right)\),

\(3C - E\)is not defined;

\(CB = \left( {\begin{aligned}{*{20}{c}}9&{ - 13}&{ - 5}\\{ - 13}&6&{ - 5}\end{aligned}} \right)\)and

\(EB\) is not defined.

Step-by-step solution

Find the matrix \(A + 2B\)

The value of \(A + 2B\) can be calculated as follows:

\(\begin{aligned}{c}A + 2B = \left( {\begin{aligned}{*{20}{c}}2&0&{ - 1}\\4&{ - 5}&2\end{aligned}} \right) + 2\left( {\begin{aligned}{*{20}{c}}7&{ - 5}&1\\1&{ - 4}&{ - 3}\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}2&0&{ - 1}\\4&{ - 5}&2\end{aligned}} \right) + \left( {\begin{aligned}{*{20}{c}}{14}&{ - 10}&2\\2&{ - 8}&{ - 6}\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}{16}&{ - 10}&1\\6&{ - 13}&{ - 4}\end{aligned}} \right)\end{aligned}\)

Find the matrix \({\bf{3}}C - E\)

The value of \(3C - E\) is not defined as \(C\) has 2 columns, whereas \(E\) has 1 column.

Find the matrix \(CB\)

The value \(CB\) can be calculated as follows:

\(\begin{aligned}{c}CB = \left( {\begin{aligned}{*{20}{c}}1&2\\{ - 2}&1\end{aligned}} \right) \times \left( {\begin{aligned}{*{20}{c}}7&{ - 5}&1\\1&{ - 4}&{ - 3}\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}{1 \times 7 + 2 \times 1}&{1 \times \left( { - 5} \right) + 2 \times \left( { - 4} \right)}&{1 \times 1 + 2 \times \left( { - 3} \right)}\\{\left( { - 2} \right) \times 7 + 1 \times 1}&{\left( { - 2} \right) \times \left( { - 5} \right) + 1 \times \left( { - 4} \right)}&{\left( { - 2} \right) \times 1 + 1 \times \left( { - 3} \right)}\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}9&{ - 13}&{ - 5}\\{ - 13}&6&{ - 5}\end{aligned}} \right)\end{aligned}\)

Find the matrix \(EB\)

The product \(EB\) is not defined as the number of columns in \(E\) does not match the number of rows of \(B\).

So, \(A + 2B = \left( {\begin{aligned}{*{20}{c}}{16}&{ - 10}&1\\6&{ - 13}&{ - 4}\end{aligned}} \right)\) and \(3C - E\) are not defined; \(CB = \left( {\begin{aligned}{*{20}{c}}9&{ - 13}&{ - 5}\\{ - 13}&6&{ - 5}\end{aligned}} \right)\) and \(EB\) are not defined.