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Chapter 2: Q12Q (page 93)

In exercise 11 and 12, the matrices are all \(n \times n\). Each part of the exercise is an implication of the form 鈥淚f 鈥渟tatement 1鈥� then 鈥渟tatement 2鈥�.鈥滿ark the implication as True if the truth of 鈥渟tatement 2鈥漚lways follows whenever 鈥渟tatement 1鈥� happens to be true. An implication is False if there is an instance in which 鈥渟tatement 2鈥� is false but 鈥渟tatement 2鈥� is false but 鈥渟tatement 1鈥� is true. Justify each answer.

a. If there is a \(n \times n\) matrix \(D\) such that \(AD = I\), then there is also an \(n \times n\) matrix \(C\) such that \(CA = I\).

b. If the columns of \(A\) are linearly independent, then the olumn of \(A\) span \({\mathbb{R}^n}\).

c. If the equation \(A{\bf{x}} = {\bf{b}}\) has at least one solution for each \({\bf{b}}\) in \({\mathbb{R}^n}\), then solution is unique for each \({\bf{b}}\).

d. If the linear transformation \(\left( x \right) \mapsto A{\bf{x}}\) maps \({\mathbb{R}^n}\) into \({\mathbb{R}^n}\), then \(A\) has \(n\) pivot position.

e. If there is a \({\bf{b}}\) in \({\mathbb{R}^n}\) such that the equation \(A{\bf{x}} = {\bf{b}}\) is inconsistent, then the transformation \(\left( x \right) \mapsto A{\bf{x}}\) is not one-to-one.

Short Answer

Expert verified

a. True

b. True

c. True

d. False

e. True

Step by step solution

01

Solve statement (a)

By the invertible matrix theorem, both the given statements are true.

Therefore, the implication is true.

02

Solve statement (b)

According to the invertible matrix theorem, if the columns are linearly independent, then the columns of A span \({\mathbb{R}^n}\).

As both the statements are true, the given implication is true.

03

Solve statement (c)

According to the inverse matrix theorem, if \(A{\bf{x}} = {\bf{b}}\) has at least one solution for each \({\rm{b}}\) in \({\mathbb{R}^n}\), then A has n pivot positions.

By the invertible matrix theorem, both statements are true. Therefore, the implication is true.

04

Solve statement (d)

By the invertible matrix theorem, if the linear transformation \({\bf{x}} \mapsto A{\bf{x}}\) maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\), then A has n pivot positions.

For the statement to be true, the mapping must be onto.

Therefore, the given implication is false.

05

Solve statement (e)

If there is a \(b\) in \({\mathbb{R}^n}\) such that the equation \(A{\bf{x}} = {\bf{b}}\) is inconsistent, then the transformation \(x \mapsto A{\bf{x}}\) is not one-to-one.

If any \(b\) in \({\mathbb{R}^n}\) is inconsistent, then the \(n \times n\) matrix is not onto and contains less than n pivots. So, the columns are linearly independent, and the transformation is not one-to-one.

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Most popular questions from this chapter

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

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Suppose AB = AC, where Band Care \(n \times p\) matrices and A is invertible. Show that B = C. Is this true, in general, when A is not invertible.

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.

37. Construct a random \({\bf{4}} \times {\bf{4}}\) matrix Aand test whether \(\left( {A + I} \right)\left( {A - I} \right) = {A^2} - I\). The best way to do this is to compute \(\left( {A + I} \right)\left( {A - I} \right) - \left( {{A^2} - I} \right)\) and verify that this difference is the zero matrix. Do this for three random matrices. Then test \(\left( {A + B} \right)\left( {A - B} \right) = {A^2} - {B^{\bf{2}}}\) the same way for three pairs of random \({\bf{4}} \times {\bf{4}}\) matrices. Report your conclusions.

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