Challenge Problems for Chapter 7

Consider the matrix \begin{equation*} A = \left [ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos t & -\sin t \\ 0 & \sin t & \cos t \end{array} \right ] \end{equation*} Does there exist a value of for which this matrix fails to have an inverse? Explain.

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No. It has a nonzero determinant for all .

If and are each matrices and is invertible, show why each of and are invertible.

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This follows because and if this product is nonzero, then each determinant in the product is nonzero and so each of these matrices is invertible.

Suppose are matrices and that Show that then
First explain why are both nonzero. Then and then show Now use what is given to conclude
Suppose is an upper triangular matrix. Show that exists if and only if all elements of the main diagonal are non zero. Is it true that will also be upper triangular? Explain. Could the same be concluded for lower triangular matrices?
The given condition is what it takes for the determinant to be non zero. Recall that the determinant of an upper triangular matrix is just the product of the entries on the main diagonal.
Let , , and denote matrices. Assume that , , and . Evaluate
(a)
(b)
(c)
If and are matrices such that , and if is odd, show that either or has no inverse.
Show that no matrix exists such that . Find a matrix with this property.
Show that for any matrices and .

Bibliography

These problems come from Chapter 3 of Ken Kuttler’s A First Course in Linear Algebra. (CC-BY)

Ken Kuttler, A First Course in Linear Algebra, Lyryx 2017, Open Edition, pp. 272–315.

Several problems came from the end of Chapter 3 of Keith Nicholson’s Linear Algebra with Applications. (CC-BY-NC-SA)

W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2018, Open Edition, pp. 167.

2024-09-06 23:46:28