- Let m < n and let A be an m×n matrix. Show that A is not one to one. Hint: Consider
the n × n matrix A
_{1}which is of the formwhere the 0 denotes an

×n matrix of zeros. Thus detA_{1}= 0 and so A_{1}is not one to one. Now observe that A_{1}x is the vector,which equals zero if and only if Ax = 0.

- Let v
_{1},,v_{n}be vectors in F^{n}and let Mdenote the matrix whose i^{th}column equals v_{i}. DefineProve that d is linear in each variable, (multilinear), that

(3.20) and

(3.21) where here e

_{j}is the vector in F^{n}which has a zero in every position except the j^{th}position in which it has a one. - Suppose f : F
^{n}×× F^{n}→ F satisfies 3.20 and 3.21 and is linear in each variable. Show that f = d. - Show that if you replace a row (column) of an n × n matrix A with itself added to some multiple of another row (column) then the new matrix has the same determinant as the original one.
- Use the result of Problem 4 to evaluate by hand the determinant
- Find the inverse if it exists of the matrix
- 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 an 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. - 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 A is det. - Suppose the characteristic polynomial of an n × n matrix A is of the form
and that a

_{0}≠0. Find a formula A^{−1}in terms of powers of the matrix A. Show that A^{−1}exists if and only if a_{0}≠0. In fact, show that a_{0}=^{n}det. - ↑Letting pdenote the characteristic polynomial of A, show that p
_{ε}≡ pis the characteristic polynomial of A + εI. Then show that if det= 0 , it follows that det≠0 wheneveris sufficiently small. - In constitutive modeling of the stress and strain tensors, one sometimes considers sums of
the form ∑
_{k=0}^{∞}a_{k}A^{k}where A is a 3×3 matrix. Show using the Cayley Hamilton theorem that if such a thing makes any sense, you can always obtain it as a finite sum having no more than n terms. - Recall you can find the determinant from expanding along the j
^{th}column.Think of det

as a function of the entries, A_{ij}. Explain why the ij^{th}cofactor is really just - Let U be an open set in ℝ
^{n}and let g :U → ℝ^{n}be such that all the first partial derivatives of all components of g exist and are continuous. Under these conditions form the matrix Dggiven byThe best kept secret in calculus courses is that the linear transformation determined by this matrix Dg

is called the derivative of g and is the correct generalization of the concept of derivative of a function of one variable. Suppose the second partial derivatives also exist and are continuous. Then show that ∑_{j}_{ij,j}= 0. Hint: First explain why ∑_{i}g_{i,k}cof_{ij}= δ_{jk}det. Next differentiate with respect to x_{j}and sum on j using the equality of mixed partial derivatives. Assume det≠0 to prove the identity in this special case. Then explain using Problem 10 why there exists a sequence ε_{k}→ 0 such that for g_{εk}≡ g+ ε_{k}x, det≠0 and so the identity holds for g_{εk}. Then take a limit to get the desired result in general. This is an extremely important identity which has surprising implications. One can build degree theory on it for example. It also leads to simple proofs of the Brouwer fixed point theorem from topology. See Evans [9] for example. - A determinant of the form
is called a Vandermonde determinant. Show it equals ∏

_{0≤i<j≤n}. By this is meant to take the product of all terms of the formsuch that j > i. Hint: Show it works if n = 1 so you are looking at. Then suppose it holds for n − 1 and consider the case n. Consider the polynomial in t,pwhich is obtained from the above by replacing the last column with the column^{T}. Explain why p= 0 for i = 0,,n − 1. Explain why p= c∏_{i=0}^{n−1}. Of course c is the coefficient of t^{n}. Find this coefficient from the above description of pand the induction hypothesis. Then plug in t = a_{n}and observe you have the formula valid for n. - The example in this exercise was shown to me by Marc van Leeuwen and it helped to correct
a misleading proof of the Cayley Hamilton theorem presented in this chapter. If
p= qfor all λ or for all λ large enough where p,qare polynomials having matrix coefficients, then it is not necessarily the case that p= qfor A a matrix of an appropriate size. The proof in question read as though it was using this incorrect argument. Let
Show that for all λ,

=I =. However,≠. Explain why this can happen. In the proof of the Cayley-Hamilton theorem given in the chapter, show that the matrix A does commute with the matrices C_{i}in that argument. Hint: Multiply both sides out with N in place of λ. Does N commute with E_{i}? - Explain how 3.19 follows from 3.18. Hint: If you have two real or complex polynomials
p,qof degree p and they are equal, for all t≠0, then by continuity, they are equal for all t. Also
thus the determinant of the one on the left equals t

^{m}det. - Explain why the proof of the Cayley-Hamilton theorem given in this chapter cannot possibly hold for arbitrary fields of scalars.
- Suppose A is m × n and B is n × m. Letting I be the identity of the appropriate size, is it
the case that det= det? Explain why or why not.

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