- (a)
- (b)
- (c)
Additional Exercises for Chapter 7: Determinants
Let . When doing cofactor expansion along the top row, we encounter three minor
matrices. What are they?
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Find the determinants of the following matrices.
- (a)
- (b)
- (c)
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- (a)
- The answer is .
- (b)
- The answer is .
- (c)
- The answer is .
Find the following determinant by (a) expanding along the first row, (b) second
column.
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Find the following determinant by (a) expanding along the first column, (b) third
row.
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Find the following determinant by expanding (a) along the second row, (b) first
column.
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Compute the determinant by cofactor expansion. Pick the easiest row or column to
use.
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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.
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It does not change the determinant. This was just taking the transpose.
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.
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In this case two rows were switched and so the resulting determinant is times the first.
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.
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The determinant is unchanged. It was just the first row added to the second.
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.
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The second row was multiplied by 2 so the determinant of the result is 2 times the original determinant.
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.
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In this case the two columns were switched so the determinant of the second is times the determinant of the first.
Let be an matrix and suppose there are rows (columns) such that all rows
(columns) are linear combinations of these rows (columns). Show
Show for an matrix and scalar .
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The matrix which has down the main diagonal has determinant equal to .
Is it true that If this is so, explain why. If it is not so, give a counter example.
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This is not true at all. Consider
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 ?
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It must be because
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?
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You would need and so or .
Let and be two matrices. ( is similar to ) means there exists an invertible matrix
such that Show that if then .
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.
Tell whether each statement is true or false. If true, provide a proof. If false, provide
a counter example.
- (a)
- If is a matrix with a zero determinant, then one column must be a multiple of some other column.
- (b)
- If any two columns of a square matrix are equal, then the determinant of the matrix equals zero.
- (c)
- For two matrices and ,
- (d)
- For an matrix ,
- (e)
- If exists then
- (f)
- If is obtained by multiplying a single row of by then
- (g)
- For an matrix,
- (h)
- If is a real matrix, then
- (i)
- If for some positive integer then
- (j)
- If for some then
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- (a)
- False. Consider
- (b)
- True.
- (c)
- False.
- (d)
- False.
- (e)
- True.
- (f)
- True.
- (g)
- True.
- (h)
- True.
- (i)
- True.
- (j)
- True.
Find the determinant using row operations to first simplify.
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One can row reduce this using only row operation 3 to and therefore, the determinant is
Find the determinant using row operations to first simplify.
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One can row reduce this using only row operation 3 to
Thus the determinant is given by
Let
Determine whether the matrix has an inverse by finding whether the determinant is
non zero. If the determinant is nonzero, find the inverse using the formula for the
inverse which involves the cofactor matrix.
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and so it has an inverse. This inverse is
Let
Determine whether the matrix has an inverse by finding whether the determinant is
non zero. If the determinant is nonzero, find the inverse using the formula for the
inverse.
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so it has an inverse. This inverse is
Let
Determine whether the matrix has an inverse by finding whether the determinant is
non zero. If the determinant is nonzero, find the inverse using the formula for the
inverse.
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so it has an inverse which is
Let
Determine whether the matrix has an inverse by finding whether the determinant is
non zero. If the determinant is nonzero, find the inverse using the formula for the
inverse.
Let
Determine whether the matrix has an inverse by finding whether the determinant is
non zero. If the determinant is nonzero, find the inverse using the formula for the
inverse.
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and so it has an inverse. The inverse turns out to equal
For the following matrices, determine if they are invertible. If so, use the formula for
the inverse in terms of the cofactor matrix to find each inverse. If the inverse does not
exist, explain why.
- (a)
- (b)
- (c)
Challenge Exercises
Consider the matrix
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
Consider the matrix
Does there exist a value of for which this matrix fails to have an inverse?
Explain.
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and so it has no inverse when
Consider the matrix
Does there exist a value of for which this matrix fails to have an inverse?
Explain.
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and so this matrix fails to have a nonzero determinant at any value of .
Consider the matrix
Does there exist a value of for which this matrix fails to have an inverse?
Explain.
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and so this matrix is always invertible.
Suppose are matrices and that Show that then
First explain why are both
nonzero. Then and then show From this use what is given to conclude Then use
Problem prb:7.36.
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You have . Hence both and have inverses. Letting be given, and so it follows from the above problem that Since is arbitrary, it follows that
Use the formula for the inverse in terms of the cofactor matrix to find the inverse of
the matrix
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Hence the inverse is
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?
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.
Practice Problem Source
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.