- Find the determinants of the following matrices.
(a)

(The answer is 31.) (b)(The answer is 375.) (c), (The answer is −2.) - Find the following determinant by expanding along the first row and second column.
- Find the following determinant by expanding along the first column and third row.
- Find the following determinant by expanding along the second row and first column.
- Compute the determinant by cofactor expansion. Pick the easiest row or column to use.
- Find the determinant using row operations.
- Find the determinant using row operations.
- Find the determinant using row operations.
- Find the determinant using row operations.
- Verify an example of each property of determinants found in Theorems 21.1.23 - 21.1.25 for 2 × 2 matrices.
- 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.
- 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.
- 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.
- 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.
- 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.
- Let A be an r ×r matrix and suppose there are r − 1 rows (columns) such that all rows (columns) are
linear combinations of these r − 1 rows (columns). Show det= 0 .
- Show det= a
^{n}detwhere here A is an n × n matrix and a is a scalar. - Illustrate with an example of 2 × 2 matrices that the determinant of a product equals the product of the determinants.
- Is it true that det= det+ det? If this is so, explain why it is so and if it is not so, give a counter example.
- An n × n matrix is called nilpotent if for some positive integer, k it follows A
^{k}= 0. If A is a nilpotent matrix and k is the smallest possible integer such that A^{k}= 0, what are the possible values of det? - A matrix is said to be orthogonal if A
^{T}A = I. Thus the inverse of an orthogonal matrix is just its transpose. What are the possible values of detif A is an orthogonal matrix? - Fill in the missing entries to make the matrix orthogonal as in Problem 21.
- Let A and B be two n × n matrices. A ∼ B (A is similar to B) means there exists an invertible
matrix S such that A = S
^{−1}BS. Show that if A ∼ B, then B ∼ A. Show also that A ∼ A and that if A ∼ B and B ∼ C, then A ∼ C. - In the context of Problem 23 show that if A ∼ B, then det= det.
- Two n × n matrices, A and B, are similar if B = S
^{−1}AS for some invertible n × n matrix S. Show that if two matrices are similar, they have the same characteristic polynomials. The characteristic polynomial of an n × n matrix M is the polynomial, det. - Tell whether the statement is true or false.
- If A is a 3 × 3 matrix with a zero determinant, then one column must be a multiple of some other column.
- If any two columns of a square matrix are equal, then the determinant of the matrix equals zero.
- For A and B two n × n matrices, det= det+ det.
- For A an n × n matrix, det= 3det
- If A
^{−1}exists then det= det^{−1}. - If B is obtained by multiplying a single row of A by 4 then det= 4det.
- For A an n × n matrix, det=
^{n}det. - If A is a real n × n matrix, then det≥ 0.
- Cramer’s rule is useful for finding solutions to systems of linear equations in which there is an infinite set of solutions.
- If A
^{k}= 0 for some positive integer, k, then det= 0 . - If Ax = 0 for some x≠0, then det= 0 .

- Use Cramer’s rule to find the solution to x + 2y = 1,2x − y = 2.
- Use Cramer’s rule to find the solution to x + 2y + z = 1,2x − y − z = 2, x + z = 1.
- Here is a matrix,
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.

- Here is a matrix,
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.

- Here is a matrix,
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.

- Here is a matrix,
- Here is a matrix,
- Use the formula for the inverse in terms of the cofactor matrix to find if possible the inverses of the
matrices
If the inverse does not exist, explain why.

- Here is a matrix,
Does there exist a value of t for which this matrix fails to have an inverse? Explain.

- Here is a matrix,
Does there exist a value of t for which this matrix fails to have an inverse? Explain.

- Here is a matrix,
Does there exist a value of t for which this matrix fails to have an inverse? Explain.

- Show that if det≠0 for A an n × n matrix, it follows that if Ax = 0, then x = 0.
- Suppose A,B are n × n matrices and that AB = I. Show that then BA = I. Hint: You might do
something like this: First explain why det,detare both nonzero. ThenA = A and then show BA= 0 . From this use what is given to conclude A= 0 . Then use Problem 38.
- Use the formula for the inverse in terms of the cofactor matrix to find the inverse of the
matrix
- Find the inverse if it exists of the matrix
- Here is a matrix,
Does there exist a value of t for which this matrix fails to have an inverse? Explain.

- Suppose A is an upper triangular matrix. Show that A
^{−1}exists if and only if all elements of the main diagonal are non zero. Is it true that A^{−1}will also be upper triangular? Explain. Is everything the same for lower triangular matrices? - If A,B, and C are each n × n matrices and ABC is invertible, why are each of A,B, and C invertible.
- Let F= det. Verify F
^{′}= det+ det. Now supposeUse Laplace expansion and the first part to verify F

^{′}=Conjecture a general result valid for n × n matrices and explain why it will be true. Can a similar thing be done with the columns?

- Let Ly = y
^{(n) }+ a_{n−1}y^{(n− 1) }++ a_{1}y^{′}+ a_{0}y where the a_{i}are given continuous functions defined on a closed interval,and y is some function which has n derivatives so it makes sense to write Ly. Suppose Ly_{k}= 0 for k = 1,2,,n. The Wronskian of these functions, y_{i}is defined asShow that for W

= Wto save space,Now use the differential equation, Ly = 0 which is satisfied by each of these functions, y

_{i}and properties of determinants presented above to verify that W^{′}+ a_{n−1}W = 0. Give an explicit solution of this linear differential equation, Abel’s formula, and use your answer to verify that the Wronskian of these solutions to the equation, Ly = 0 either vanishes identically onor never. Hint: To solve the differential equation, let A^{′}= a_{n−1}and multiply both sides of the differential equation by e^{A(x) }and then argue the left side is the derivative of something. - Find the following determinants and the inverses of the given matrices. You might use MATLAB to
do this with no trouble.
(a) det

(b) det - Find the eigenvalues and eigenvectors of the following matrices. List the eigenvalues according to
multiplicity as a root of the characteristic polyinomial.
- The eigenspace for an eigenvalue λ is defined to be the span of all eigenvectors. If the dimension of the eigenspace for each λ equals the multiplicity of the eigenvalue as a root of the characteristic polynomial, then the matrix is said to be nondefective. If, for any eigenvalue, the dimension of the eigenspace called geometric multiplicity is less than the algebraic multiplicity of the eigenvalue as a root of the characteristic polynomial, then the matrix is called defective. It can be shown that A can be diagonalized if and only if it is nondefective. See Theorem 7.5.3.
- The typical situation is that an n × n matrix has n distinct eigenvalues. In this case, the matrix is
always nondefective. This comes from the following theorem which you will show in this
problem.
Theorem 21.5.1 Let A be an n × n matrix and let

be distinct eigenvalues corresponding to eigenvectors. Then this set of eigenvectors is a linearly independent set.Do the following. If not independent, then there exist scalars a

_{i}such thatin which the a

_{i}are not all zero and l is as small as possible for this to take place. Explain why a_{l}≠0 and why l ≥ 2. Then multiply both sides on the left by A and then both sides on the left by μ_{l}. Subtract and obtain a contradiction of some sort, having to do with l being as small as possible and all eigenvectors being nonzero.

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