Challenge Problems for Chapter 4

Solve for the matrix if:
(a)
;
(b)
;

where , ,
,

Consider \begin{equation*} p(X) = X^{3} - 5X^{2} + 11X - 4I. \end{equation*}
(a)
If compute .
(b)
If where is , find in terms of .

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.

Assume that a system of linear equations has at least two distinct solutions and .
(a)
Show that is a solution for every .
(b)
Show that implies .
(c)
Deduce that has infinitely many solutions.

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(a)
If , then . So . But is not zero (because and are distinct), so .
Let
(a)
show that .
(b)
What is wrong with the following argument? If , then , so , whence or .
Let and be elementary matrices obtained from the identity matrix by adding multiples of row to rows and . If and , show that .

Bibliography

These exercises come from the end of Chapter 2 of Keith Nicholson’s Linear Algebra with Applications. (CC-BY-NC-SA)

W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2018, Open Edition, pp. 143–144.

2024-09-06 23:46:47