A First Course in Linear Algebra, 2017a

70 Matrices
Definition 2.29: Symmetric and Skew Symmetric Matrices
An n × n matrix A is said to be symmetric if A = A T . It is said to be skew symmetric if A = −A T .
We will explore these definitions in the following examples.
Example 2.30: Symmetric Matrices
Let
⎡
A = ⎣
Use Definition 2.29 to show that A is symmetric.
2 1 3
1 5 −3
3 −3 7
⎤
⎦
Solution. By Definition 2.29, we need to show that A = A T . Now, using Definition 2.26,
⎡
2 1
⎤
3
A T = ⎣ 1 5 −3 ⎦
3 −3 7
Hence, A = A T ,soA is symmetric.
♠
Example 2.31: A Skew Symmetric Matrix
Let
Show that A is skew symmetric.
⎡
A = ⎣
0 1 3
−1 0 2
−3 −2 0
⎤
⎦
Solution. By Definition 2.29,
⎡
A T = ⎣
0 −1 −3
1 0 −2
3 2 0
⎤
⎦
You can see that each entry of A T is equal to −1 times the same entry of A. Hence, A T = −A and so
by Definition 2.29, A is skew symmetric.
♠