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Chapter 2: Q2Q (page 93)

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

Expert verified

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

\(3C - E\)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

01

Find the matrix \(A + 2B\)

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

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

02

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.

03

Find the matrix \(CB\)

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

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

04

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.

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Most popular questions from this chapter

Suppose a linear transformation \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) has the property that \(T\left( {\mathop{\rm u}\nolimits} \right) = T\left( {\mathop{\rm v}\nolimits} \right)\) for some pair of distinct vectors u and v in \({\mathbb{R}^n}\). Can Tmap \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\)? Why or why not?

In Exercises 1–9, assume that the matrices are partitioned conformably for block multiplication. In Exercises 5–8, find formulas for X, Y, and Zin terms of A, B, and C, and justify your calculations. In some cases, you may need to make assumptions about the size of a matrix in order to produce a formula. [Hint:Compute the product on the left, and set it equal to the right side.]

6. \[\left[ {\begin{array}{*{20}{c}}X&{\bf{0}}\\Y&Z\end{array}} \right]\left[ {\begin{array}{*{20}{c}}A&{\bf{0}}\\B&C\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}I&{\bf{0}}\\{\bf{0}}&I\end{array}} \right]\]

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?

Describe in words what happens when you compute \({A^{\bf{5}}}\), \({A^{{\bf{10}}}}\), \({A^{{\bf{20}}}}\), and \({A^{{\bf{30}}}}\) for \(A = \left( {\begin{aligned}{*{20}{c}}{1/6}&{1/2}&{1/3}\\{1/2}&{1/4}&{1/4}\\{1/3}&{1/4}&{5/12}\end{aligned}} \right)\).

Let \(X\) be \(m \times n\) data matrix such that \({X^T}X\) is invertible., and let \(M = {I_m} - X{\left( {{X^T}X} \right)^{ - {\bf{1}}}}{X^T}\). Add a column \({x_{\bf{0}}}\) to the data and form

\(W = \left[ {\begin{array}{*{20}{c}}X&{{x_{\bf{0}}}}\end{array}} \right]\)

Compute \({W^T}W\). The \(\left( {{\bf{1}},{\bf{1}}} \right)\) entry is \({X^T}X\). Show that the Schur complement (Exercise 15) of \({X^T}X\) can be written in the form \({\bf{x}}_{\bf{0}}^TM{{\bf{x}}_{\bf{0}}}\). It can be shown that the quantity \({\left( {{\bf{x}}_{\bf{0}}^TM{{\bf{x}}_{\bf{0}}}} \right)^{ - {\bf{1}}}}\) is the \(\left( {{\bf{2}},{\bf{2}}} \right)\)-entry in \({\left( {{W^T}W} \right)^{ - {\bf{1}}}}\). This entry has a useful statistical interpretation, under appropriate hypotheses.

In the study of engineering control of physical systems, a standard set of differential equations is transformed by Laplace transforms into the following system of linear equations:

\(\left[ {\begin{array}{*{20}{c}}{A - s{I_n}}&B\\C&{{I_m}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{\bf{x}}\\{\bf{u}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{y}}\end{array}} \right]\)

Where \(A\) is \(n \times n\), \(B\) is \(n \times m\), \(C\) is \(m \times n\), and \(s\) is a variable. The vector \({\bf{u}}\) in \({\mathbb{R}^m}\) is the “input” to the system, \({\bf{y}}\) in \({\mathbb{R}^m}\) is the “output” and \({\bf{x}}\) in \({\mathbb{R}^n}\) is the “state” vector. (Actually, the vectors \({\bf{x}}\), \({\bf{u}}\) and \({\bf{v}}\) are functions of \(s\), but we suppress this fact because it does not affect the algebraic calculations in Exercises 19 and 20.)
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