Click the arrow to see answer.
Solved Problems for Chapter 7
Let . When doing cofactor expansion along the top row, we encounter three minor
matrices. What are they?
Find the following determinant by (a) expanding along the first row, (b) second
column.
Click the arrow to see answer.
Compute the determinant by co-factor expansion. Pick the easiest row or column to
use.
Click the arrow to see answer.
An operation is done to get from the first matrix to the second. Identify
what was done and tell how it will affect the value of the determinant.
Click the arrow to see answer.
An operation is done to get from the first matrix to the second. Identify
what was done and tell how it will affect the value of the determinant.
Click the arrow to see answer.
An operation is done to get from the first matrix to the second. Identify
what was done and tell how it will affect the value of the determinant.
Click the arrow to see answer.
An operation is done to get from the first matrix to the second. Identify
what was done and tell how it will affect the value of the determinant.
Click the arrow to see answer.
An operation is done to get from the first matrix to the second. Identify
what was done and tell how it will affect the value of the determinant.
Click the arrow to see answer.
Let be an matrix and suppose there are rows such that all rows are linear
combinations of these rows. Show .
Click the arrow to see answer.
An matrix is called nilpotent if for some positive integer, it follows If is a
nilpotent matrix and is the smallest possible integer such that what are the possible
values of ?
Click the arrow to see answer.
A matrix is said to be orthogonal if Thus the inverse of an orthogonal matrix
is just its transpose. What are the possible values of if is an orthogonal
matrix?
Click the arrow to see answer.
Let and be two matrices. We say that is similar to and write provided that
there exists an invertible matrix such that Show that if then .
Click the arrow to see answer.
Determine whether each statement is true or false. If true, provide a proof. If false,
provide a counter example.
- (a)
- If any two columns of a square matrix are equal, then the determinant of the matrix equals zero.
- (b)
- If exists then
- (c)
- If is a real matrix, then
- (d)
- If for some then
Click the arrow to see answer.
Let
Determine whether the matrix has an inverse by finding whether the determinant is
non-zero.
Click the arrow to see answer.
Bibliography
Some of the problems come from the end of 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.
2024-09-06 23:25:48