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Chapter 2: Q27Q (page 93)

Exercises 27 and 28 prove special cases of the facts about elementary matrices stated in the box following Example 5. Here Ais a \({\bf{3}} \times {\bf{3}}\) matrix and \(I = {I_{\bf{3}}}\). (A general proof would require slightly more notation.)

27. a. Use equation (1) from Section 2.1 to show that \(ro{w_i}\left( A \right) = ro{w_i}\left( I \right) \cdot A\) for \(i = 1,2,3\).

b. Show that if rows 1 and 2 of Aare interchanged, then the result may be written as EA, where Eis an elementary matrix formed by interchanging rows 1 and 2 of I.

c. Show that if row 3 of Ais multiplied by 5, then the result may be written as EA, where Eis formed by multiplying row 3 of Iby 5.

Short Answer

Expert verified
  1. It is proved that\({\rm{ro}}{{\rm{w}}_i}\left( A \right) = {\rm{ro}}{{\rm{w}}_i}\left( I \right) \cdot A\).
  1. If rows one and two of Aare interchanged, the result can be written as EA.
  1. If row three of Ais multiplied by 5, the result can be written as EA.

Step by step solution

01

(a) Step 1: Use the row-column rule

Consider the equation from section 2.1 as shown below:

\({\rm{ro}}{{\rm{w}}_i}\left( {BA} \right) = {\rm{ro}}{{\rm{w}}_i}\left( B \right) \cdot A\)

Here, the\(i{\rm{ th}}\)row of matrix A is\({\rm{ro}}{{\rm{w}}_i}\left( A \right)\).

Replace matrix B with theidentity matrix I as shown below:

Thus, it is proved that .

02

(b) Step 2: Interchange the rows

Suppose for, matrix A is .

Interchange rows one and two as shown below:

Apply the equation.

Hence, proved.

03

(c) Step 3: Apply row operations

Suppose for, matrix A is .

Multiply row three by 5 as shown below:

Apply the equation.

Hence, proved.

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Most popular questions from this chapter

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

Suppose Ais a \(3 \times n\) matrix whose columns span \({\mathbb{R}^3}\). Explain how to construct an \(n \times 3\) matrix Dsuch that \(AD = {I_3}\).

A useful way to test new ideas in matrix algebra, or to make conjectures, is to make calculations with matrices selected at random. Checking a property for a few matrices does not prove that the property holds in general, but it makes the property more believable. Also, if the property is actually false, you may discover this when you make a few calculations.

38. Use at least three pairs of random \(4 \times 4\) matrices Aand Bto test the equalities \({\left( {A + B} \right)^T} = {A^T} + {B^T}\) and \({\left( {AB} \right)^T} = {A^T}{B^T}\). (See Exercise 37.) Report your conclusions. (Note:Most matrix programs use \(A'\) for \({A^{\bf{T}}}\).

Let \(A = \left( {\begin{aligned}{*{20}{c}}{\bf{1}}&{\bf{2}}\\{\bf{5}}&{{\bf{12}}}\end{aligned}} \right),{b_{\bf{1}}} = \left( {\begin{aligned}{*{20}{c}}{ - {\bf{1}}}\\{\bf{3}}\end{aligned}} \right),{b_{\bf{2}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{1}}\\{ - {\bf{5}}}\end{aligned}} \right),{b_{\bf{3}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{2}}\\{\bf{6}}\end{aligned}} \right),\) and \({b_{\bf{4}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{3}}\\{\bf{5}}\end{aligned}} \right)\).

  1. Find \({A^{ - {\bf{1}}}}\), and use it to solve the four equations \(Ax = {b_{\bf{1}}},\)\(Ax = {b_2},\)\(Ax = {b_{\bf{3}}},\)\(Ax = {b_{\bf{4}}}\)\(\)
  2. The four equations in part (a) can be solved by the same set of row operations, since the coefficient matrix is the same in each case. Solve the four equations in part (a) by row reducing the augmented matrix \(\left( {\begin{aligned}{*{20}{c}}A&{{b_{\bf{1}}}}&{{b_{\bf{2}}}}&{{b_{\bf{3}}}}&{{b_{\bf{4}}}}\end{aligned}} \right)\).

Let \(A = \left( {\begin{aligned}{*{20}{c}}{\bf{2}}&{ - {\bf{3}}}\\{ - {\bf{4}}}&{\bf{6}}\end{aligned}} \right)\) and \(B = \left( {\begin{aligned}{*{20}{c}}{\bf{8}}&{\bf{4}}\\{\bf{5}}&{\bf{5}}\end{aligned}} \right)\) and \(C = \left( {\begin{aligned}{*{20}{c}}{\bf{5}}&{ - {\bf{2}}}\\{\bf{3}}&{\bf{1}}\end{aligned}} \right)\). Verfiy that \(AB = AC\) and yet \(B \ne C\).
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