Math 2270
Quiz 11
1. Use Gaussian elimination to find the determinant of

1 3
 1 6
A=
 1 3
2 6
First, we row reduce A to upper triangular

1 3
 1 6

 1 3
2 6

2 4
4 8 
.
0 0 
4 12
form.
2
4
0
4
 
1
4
 0
8 
∼
0   0
0
12
3
3
0
0

2
4
2
4 
.
−2 −4 
0
4
That was pretty easy. Since we performed no row swaps and divided no rows by constants, we have that
det(A) = 1 · 3 · −2 · 4 = −24.
2. Consider a skew symmetric n × n matrix A, where n is odd. Show that A is noninvertible by showing that
det(A) = 0.
If A is skew symmetric, then AT = −A. Taking the determinant of both sides, we have that
det(AT ) = det(−A).
But det(AT ) = det(A) and, as was discussed in class, det(−A) = (−1)n det(A) = −det(A), since n is odd. This
means that
det(A) = −det(A),
and the only number that is equal to its opposite is zero.