- Show that matrix multiplication is associative. That is,
C = A
- Show the inverse of a matrix, if it exists, is unique. Thus if AB = BA = I,
then B = A−1.
- In the proof of Theorem 4.7.14 it was claimed that det = 1
. Here I =
Prove this assertion. Also prove Corollary 4.7.17.
- Let v1,
,vn be vectors in Fn and let M denote the matrix whose
column equals vi. Define
Prove that d is linear in each variable, (multilinear), that
where here ej is the vector in Fn which has a zero in every position except the jth
position in which it has a one.
- Suppose f : Fn ×
× Fn → F satisfies 4.11.37 and 4.11.38 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.
- If A =
, show det =
- Use the result of Problem 6 to evaluate by hand the determinant
- Find the inverse if it exists of the matrix,
- Let Ly = y
y′ + a0
y where the ai 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
Lyk = 0 for k = 1,2,
,n. The Wronskian of these functions, yi is defined
Show that for W =
to save space,
Now use the differential equation, Ly = 0 which is satisfied by each of these
functions, yi and properties of determinants presented above to verify that
W′ + an−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 on or
- Two n×n matrices, A and B, are similar if B = S−1AS for some invertible n×n
matrix, S. Show that if two matrices are similar, they have the same characteristic
- Suppose the characteristic polynomial of an n × n matrix, A is of the
and that a0≠0. Find a formula A−1 in terms of powers of the matrix, A. Show that
A−1 exists if and only if a0≠0.
- In constitutive modeling of the stress and strain tensors, one sometimes considers
sums of the form ∑
k=0∞akAk 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.