MATH 520 — REVIEW FOR FINAL EXAM

(a) Find L and U so that A = LU where L is lower triangular with 1’s on the diagonal
and U is upper triangular.
(b) Solve Ax = b where b =
4. Let
Find A −1 .
5. Let
1
.
i
⎡
1 0 0
⎤
0
⎢ 2
A = ⎣
3
1
4
0
1
0 ⎥
⎦ .
0
1 1 1 1
⎡
0 0 1/4
⎤
1
⎢ 1/4
A = ⎣
1/2
0
1
3/4
0
0 ⎥
⎦ .
0
1/4 0 0 0
(a) Find k with A k > 0. Conclude that A is a regular Markov matrix.
(b) Find A ∞ = limk→∞ A k .
6. (a) Let A be a non-zero 3×3 real, anti-symmetric matrix with A 2 = D, which is
diagonal. Show that 7 of the 9 entries in A are zero.
(b) Let A be a non-zero 4×4 real, anti-symmetric matrix with A 2 = D, which is
diagonal. Show that the result in (a) does not generalize. Find such an A with each
non-diagonal entry = ±1 (and furthermore A 2 = −3I).
7. Let
⎡
0
A = ⎣ −1
1
0
⎤
1
1 ⎦ .
−1 −1 0
(a) Find the eigenvalues and an orthonormal basis of eigenvector. Show that A =
UΛU H where U is unitary and Λ is diagonal.
2