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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

Suppose Tand U are linear transformations from \({\mathbb{R}^n}\) to \({\mathbb{R}^n}\) such that \(T\left( {U{\mathop{\rm x}\nolimits} } \right) = {\mathop{\rm x}\nolimits} \) for all x in \({\mathbb{R}^n}\) . Is it true that \(U\left( {T{\mathop{\rm x}\nolimits} } \right) = {\mathop{\rm x}\nolimits} \) for all x in \({\mathbb{R}^n}\)? Why or why not?

[M] Suppose memory or size restrictions prevent your matrix program from working with matrices having more than 32 rows and 32 columns, and suppose some project involves \(50 \times 50\) matrices A and B. Describe the commands or operations of your program that accomplish the following tasks.

a. Compute \(A + B\)

b. Compute \(AB\)

c. Solve \(Ax = b\) for some vector b in \({\mathbb{R}^{50}}\), assuming that \(A\) can be partitioned into a \(2 \times 2\) block matrix \(\left[ {{A_{ij}}} \right]\), with \({A_{11}}\) an invertible \(20 \times 20\) matrix, \({A_{22}}\) an invertible \(30 \times 30\) matrix, and \({A_{12}}\) a zero matrix. [Hint: Describe appropriate smaller systems to solve, without using any matrix inverse.]

Prove Theorem 2(d). (Hint: The \(\left( {i,j} \right)\)- entry in \(\left( {rA} \right)B\) is \(\left( {r{a_{i1}}} \right){b_{1j}} + ... + \left( {r{a_{in}}} \right){b_{nj}}\).)

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

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.

36. Write the command(s) that will create a \(6 \times 4\) matrix with random entries. In what range of numbers do the entries lie? Tell how to create a \(3 \times 3\) matrix with random integer entries between \( - {\bf{9}}\) and 9. (Hint:If xis a random number such that 0 < x < 1, then \( - 9.5 < 19\left( {x - .5} \right) < 9.5\).
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