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Chapter 2: Q6Q (page 93)

In exercise 5 and 6, compute the product \(AB\) in two ways: (a) by the definition, where \(A{b_{\bf{1}}}\) and \(A{b_{\bf{2}}}\) are computed separately, and (b) by the row-column rule for computing \(AB\).

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

Short Answer

Expert verified

\(\left( {\begin{aligned}{*{20}{c}}0&{14}\\{ - 3}&{ - 9}\\{13}&4\end{aligned}} \right)\)

Step by step solution

01

Find the value of \(A{b_{\bf{1}}}\)

Multiply matrix \(A\) with the first column of matrix \(B\).

\(\begin{aligned}{c}A{b_1} = \left( {\begin{aligned}{*{20}{c}}4&{ - 2}\\{ - 3}&0\\3&5\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}1\\2\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}{4 \times 1 + \left( { - 2} \right) \times 2}\\{\left( { - 3} \right) \times 1 + 0}\\{3 \times 1 + 5 \times 2}\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}0\\{ - 3}\\{13}\end{aligned}} \right)\end{aligned}\)

02

Find the value of \(A{b_{\bf{2}}}\)

Multiply matrix \(A\) with the second column of matrix \(B\).

\(\begin{aligned}{c}A{b_2} = \left( {\begin{aligned}{*{20}{c}}4&{ - 2}\\{ - 3}&0\\3&5\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}3\\{ - 1}\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}{4 \times 3 + \left( { - 2} \right) \times \left( { - 1} \right)}\\{\left( { - 3} \right) \times 3 + 0}\\{3 \times 3 + 5 \times \left( { - 1} \right)}\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}{14}\\{ - 9}\\4\end{aligned}} \right)\end{aligned}\)

03

Write the product \(AB\)

The product \(AB\) can be written as follows:

\(\begin{aligned}{c}AB = \left( {\begin{aligned}{*{20}{c}}{A{b_1}}&{A{b_2}}\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}0&{14}\\{ - 3}&{ - 9}\\{13}&4\end{aligned}} \right)\end{aligned}\)

04

Find the product \(AB\) using row-column rule

\(\begin{aligned}{c}AB = \left( {\begin{aligned}{*{20}{c}}4&{ - 2}\\{ - 3}&0\\3&5\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}1&3\\2&{ - 1}\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}{4 \times 1 + \left( { - 2} \right) \times 2}&{4 \times 3 + \left( { - 2} \right) \times \left( { - 1} \right)}\\{\left( { - 3} \right) \times 1 + 0}&{\left( { - 3} \right) \times 3 + 0}\\{3 \times 1 + 5 \times 2}&{3 \times 3 + 5 \times \left( { - 1} \right)}\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}0&{14}\\{ - 3}&{ - 9}\\{13}&4\end{aligned}} \right)\end{aligned}\)

So, \(AB = \left( {\begin{aligned}{*{20}{c}}0&{14}\\{ - 3}&{ - 9}\\{13}&4\end{aligned}} \right)\).

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Most popular questions from this chapter

Suppose the first two columns, \({{\bf{b}}_1}\) and \({{\bf{b}}_2}\), of Bare equal. What can you say about the columns of AB(if ABis defined)? Why?

Suppose the last column of ABis entirely zero but Bitself has no column of zeros. What can you sayaboutthe columns of A?

In Exercise 9 mark each statement True or False. Justify each answer.

9. a. In order for a matrix B to be the inverse of A, both equations \(AB = I\) and \(BA = I\) must be true.

b. If A and B are \(n \times n\) and invertible, then \({A^{ - {\bf{1}}}}{B^{ - {\bf{1}}}}\) is the inverse of \(AB\).

c. If \(A = \left( {\begin{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right)\) and \(ab - cd \ne {\bf{0}}\), then A is invertible.

d. If A is an invertible \(n \times n\) matrix, then the equation \(Ax = b\) is consistent for each b in \({\mathbb{R}^{\bf{n}}}\).

e. Each elementary matrix is invertible.

Suppose block matrix \(A\) on the left side of (7) is invertible and \({A_{{\bf{11}}}}\) is invertible. Show that the Schur component \(S\) of \({A_{{\bf{11}}}}\) is invertible. [Hint: The outside factors on the right side of (7) are always invertible. Verify this.] When \(A\) and \({A_{{\bf{11}}}}\) are invertible, (7) leads to a formula for \({A^{ - {\bf{1}}}}\), using \({S^{ - {\bf{1}}}}\) \(A_{{\bf{11}}}^{ - {\bf{1}}}\), and the other entries in \(A\).

In Exercises 1鈥�9, assume that the matrices are partitioned conformably for block multiplication. Compute the products shown in Exercises 1鈥�4.

1. \(\left[ {\begin{array}{*{20}{c}}I&{\bf{0}}\\E&I\end{array}} \right]\left[ {\begin{array}{*{20}{c}}A&B\\C&D\end{array}} \right]\)
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