Mathematical Methods for Physicists: A concise introduction - Site Map

PROBLEMS
(g) Is A ~ A ~ T antisymmetric?
3.7 Show that the matrix
0
1 4
1
0
B ~A ˆ @ 2 5
C
0A
3 6 0
is not invertible.
3.8 Show that if A ~ and ~B are invertible matrices of the same order, then A ~ ~B is
invertible.
3.9 Given
0
1 2
1
3
B ~A ˆ @ 2 5
C
3A;
1 0 8
®nd ~ A 1 and check the answer by direct multiplication.
3.10 Prove that if ~A is a non-singular matrix, then det( ~A 1 †ˆ1= det… ~A).
3.11 If ~A is an invertible n n matrix, show that ~AX ˆ 0 has only the trivial
solution.
3.12 Show, by computing a matrix inverse, that the solution to the following
system is x 1 ˆ 4, x 2 ˆ 1:
x 1 x 2 ˆ 3;
x 1 ‡ x 2 ˆ 5:
3.13 Solve the system ~AX ˆ ~B if
0
1 0
1
0
0 1
1
B ~A ˆ @ 0 2
C
0A; B C ~B ˆ @ 2 A:
0 0 1
3
3.14 Given matrix ~A, ®nd A*, A T , and A y , where
0
2 ‡ 3i 1 i 5i 3
1
B ~A ˆ @ 1 ‡ i 6 i 1 ‡ 3i
C
1 2i A:
5 6i 3 0 4
3.15 Show that:
(a) The matrix ~A ~A y , where ~A is any matrix, is hermitian.
(b) … ~A ~B† y ˆ ~B y ~A y :
(c) If~A; ~B are hermitian, then ~A ~B ‡ ~B ~A is hermitian.
(d) If~A and ~B are hermitian, then i… ~A ~B ~B ~A† is hermitian.
3.16 Obtain the most general orthogonal matrix of order 2.
[Hint: use relations (3.34a) and (3.34b).]
3.17. Obtain the most general unitary matrix of order 2.
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