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Chapter 2: Q2.9-24E (page 93)

In Exercises 19-24, justify each answer or construction.

Construct a \({\bf{4}} \times {\bf{3}}\) matrix with rank 1.

Short Answer

Expert verified

\(\left[ {\begin{array}{*{20}{c}}1&3&{ - 1}\\2&6&{ - 2}\\3&9&{ - 3}\\4&{12}&{ - 4}\end{array}} \right]\)

Step by step solution

01

State the condition for a matrix of rank 1

A rank-1 matrix has one-dimensional column space. For rank1, every column vector should be a multiple of acommon vector.

02

Write an example of rank-1 matrix

An example of a rank-1 matrix is shown below:

\(\left[ {\begin{array}{*{20}{c}}1&3&{ - 1}\\2&6&{ - 2}\\3&9&{ - 3}\\4&{12}&{ - 4}\end{array}} \right]\)

So, \(\left[ {\begin{array}{*{20}{c}}1&3&{ - 1}\\2&6&{ - 2}\\3&9&{ - 3}\\4&{12}&{ - 4}\end{array}} \right]\) represents a matrix of rank 1.

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Suppose Aand Bare \(n \times n\), Bis invertible, and ABis invertible. Show that Ais invertible. (Hint: Let C=AB, and solve this equation for A.)

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