Problem #1: (a) For what values of a, b, and c is the following matrix symmetric? −3 5a − c 5a + 2b a 2 3 a + 7b c a (b) An n × n matrix A is called skew-symmetric if AT = −A. What values of a, b, c, and d now make the following matrix skew-symmetric? d 5a − c 5a + 2b a 0 3 − 2d a + 7b c 0 Problem #2: For what value of a do the following matrices commute? A =  −5a 2 −5 0 ,  B =  1 2 −5 −2 Problem #3: Find the values of a, b, and c so that det(A) = ax2 + bx + c. A =  x -2 0 -1 x -2 1 -2 -3 Problem #4: (a) Find the determinant of the following matrix. A =  -1 0 2 0 2 1 0 -3 1 0 1 -4 1 0 -3 0 (b) Find the determinant of the following matrix.  B =  4 4 0 0 0 -4 4 0 0 0 3 -4 -3 0 0 2 4 4 3 0 -4 2 2 2 -4 Hint: Do a row operation first.  Problem #5: Consider the following matrix. A =  a b c d e f g h i   Suppose that det(A) = −4. Let B be another 3 × 3 matrix (not given here) with det(B) = 4. Find the determinant of each of the following matrices. (a) C =  a + 3g -g -3d b + 3h -h -3e c + 3i -i -3f (b) D = 3A−1BT (c) E = −BA3 (d) F = adj(B) Problem #6: Consider the following matrix. A =  -2 -1 -5 2 -8 -3 3 4 1 Let B = adj(A). Find b31, b32, and b33. (i.e., find the entries in the third row of the adjoint of A.) Problem#7: Let A and B be n × n matrices. Which of the following statements are always true? (i) If det(A) = det(B) then det(A − B) = 0. (ii) If A and B are symmetric, then the matrix AB is also symmetric. (iii) If A and B are skew-symmetric, then the matrix AT + B is also skew-symmetric. (A) (i) and (ii) only   (B) none of them   (C) (ii) and (iii) only   (D) (i) only   (E) (ii) only   (F) all of them   (G) (iii) only   (H) (i) and (iii) only    Problem #8: Find the eigenvalues of the following matrix. A =  5 -2 5 0 -4 0 -1 2 -1 Problem #9: Consider the following matrix A (whose 2nd and 3rd rows are not given), and vector x. A =  2 2 -8 * * * * * * ,   x =  2 -2 -2 Given that x is an eigenvector of the matrix A, what is the corresponding eigenvalue? .

1. Problem #1:
(a)
For what values of a, b, and c is the following matrix
symmetric?
−3
5a − c
5a + 2b
a
2
3
a + 7b
c
a
(b)
An n × n matrix A is called skew-symmetric if AT = −A. What
values of a, b, c, and d now make the following matrix skew-
symmetric?
d
5a − c
5a + 2b
a
0
3 − 2d
a + 7b
c
0

2. Problem #2:
For what value of a do the following matrices commute?
A =
−5a
2
−5
0
, B =
1
2
−5
−2

3. Problem #3:
Find the values of a, b, and c so that det(A) = ax2 + bx + c.
A =
x
-2
0
-1
x
-2
1
-2
-3
Problem #4:
(a)
Find the determinant of the following matrix.
A =
-1
0
2
0
2
1

4. 0
-3
1
0
1
-4
1
0
-3
0
(b)
Find the determinant of the following matrix.
B =
4
4
0
0
0
-4
4
0
0
0
3
-4
-3
0
0
2
4

5. 4
3
0
-4
2
2
2
-4
Hint: Do a row operation first.
Problem #5:
Consider the following matrix.
A =
a
b
c
d
e
f
g

6. h
i
Suppose that det(A) = −4. Let B be another 3 × 3 matrix (not
given here) with det(B) = 4. Find the determinant of each of the
following matrices.
(a)
C =
a + 3g
-g
-3d
b + 3h
-h
-3e
c + 3i
-i
-3f
(b)
D = 3A−1BT
(c)
E = −BA3
(d)
F = adj(B)

7. Problem #6:
Consider the following matrix.
A =
-2
-1
-5
2
-8
-3
3
4
1
Let B = adj(A). Find b31, b32, and b33. (i.e., find the entries in
the third row of the adjoint of A.)

8. Problem#7:
Let A and B be n × n matrices. Which of the following
statements are always true?
(i) If det(A) = det(B) then det(A − B) = 0.
(ii) If A and B are symmetric, then the matrix AB is also
symmetric.
(iii) If A and B are skew-symmetric, then the matrix AT + B is
also skew-symmetric.
(A) (i) and (ii) only (B) none of them (C) (ii) and (iii) only
(D) (i) only (E) (ii) only (F) all of them (G) (iii) only
(H) (i) and (iii) only
Problem #8:
Find the eigenvalues of the following matrix.
A =
5
-2

9. 5
0
-4
0
-1
2
-1
Problem #9:
Consider the following matrix A (whose 2nd and 3rd rows are
not given), and vector x.
A =
2
2
-8
*
*
*
*
*
*

10. , x =
2
-2
-2
Given that x is an eigenvector of the matrix A, what is the
corresponding eigenvalue?