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Chapter 2: Q3SE (page 93)

Let \(A = \left( {\begin{aligned}{*{20}{c}}{\bf{0}}&{\bf{0}}&{\bf{0}}\\{\bf{1}}&{\bf{0}}&{\bf{0}}\\{\bf{0}}&{\bf{1}}&{\bf{0}}\end{aligned}} \right)\). Show that \({A^{\bf{3}}} = {\bf{0}}\). Use matrix algebra to complete the product \(\left( {I - A} \right)\left( {I + A + {A^{\bf{2}}}} \right)\).

Short Answer

Expert verified

The equation \({A^3} = 0\) is proved. Using \({A^3} = 0\), the product \(\left( {I - A} \right)\left( {I + A + {A^2}} \right) = I\).

Step by step solution

01

First compute \({A^{\bf{2}}}\)

\(\begin{aligned}{c}{A^2} = AA\\ = \left( {\begin{aligned}{*{20}{c}}0&0&0\\1&0&0\\0&1&0\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}0&0&0\\1&0&0\\0&1&0\end{aligned}} \right)\\{A^2} = \left( {\begin{aligned}{*{20}{c}}0&0&0\\0&0&0\\1&0&0\end{aligned}} \right)\end{aligned}\)

02

Using \({A^{\bf{2}}}\) compute \({A^{\bf{3}}}\)

\(\begin{aligned}{c}{A^3} = {A^2}A\\ = \left( {\begin{aligned}{*{20}{c}}0&0&0\\0&0&0\\1&0&0\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}0&0&0\\1&0&0\\0&1&0\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}0&0&0\\0&0&0\\0&0&0\end{aligned}} \right)\\{A^3} = 0\end{aligned}\)

03

Use matrix algebra

\(\begin{aligned}{c}\left( {I - A} \right)\left( {I + A + {A^2}} \right) = {I^2} + IA + I{A^2} - AI - {A^2} - {A^3}\\ = I + A + {A^2} - A - {A^2} - {A^3}\\ = I - {A^3}\\ = I - 0\\\left( {I - A} \right)\left( {I + A + {A^2}} \right) = I\end{aligned}\)

\({A^3} = 0\) from step 2.

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Most popular questions from this chapter

Suppose Ais an \(n \times n\) matrix with the property that the equation \(Ax = 0\)has only the trivial solution. Without using the Invertible Matrix Theorem, explain directly why the equation \(Ax = b\) must have a solution for each b in \({\mathbb{R}^n}\).

Let \(A = \left( {\begin{aligned}{*{20}{c}}3&{ - 6}\\{ - 1}&2\end{aligned}} \right)\). Construct a \({\bf{2}} \times {\bf{2}}\) matrix Bsuch that ABis the zero matrix. Use two different nonzero columns for B.

In Exercises 1鈥�9, assume that the matrices are partitioned conformably for block multiplication. In Exercises 5鈥�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.]

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

Let Ube the \({\bf{3}} \times {\bf{2}}\) cost matrix described in Example 6 of Section 1.8. The first column of Ulists the costs per dollar of output for manufacturing product B, and the second column lists the costs per dollar of output for product C. (The costs are categorized as materials, labor, and overhead.) Let \({q_1}\) be a vector in \({\mathbb{R}^{\bf{2}}}\) that lists the output (measured in dollars) of products B and C manufactured during the first quarter of the year, and let \({q_{\bf{2}}}\), \({q_{\bf{3}}}\) and \({q_{\bf{4}}}\) be the analogous vectors that list the amounts of products B and C manufactured in the second, third, and fourth quarters, respectively. Give an economic description of the data in the matrix UQ, where \(Q = \left( {\begin{aligned}{*{20}{c}}{{{\bf{q}}_1}}&{{{\bf{q}}_2}}&{{{\bf{q}}_3}}&{{{\bf{q}}_4}}\end{aligned}} \right)\).

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 \(AB\).

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