A First Course in Linear Algebra, 2017a

94 Matrices
(a) (A − B) 2 = A 2 − 2AB + B 2
(b) (AB) 2 = A 2 B 2
(c) (A + B) 2 = A 2 + 2AB + B 2
(d) (A + B) 2 = A 2 + AB + BA + B 2
(e) A 2 B 2 = A(AB)B
(f) (A + B) 3 = A 3 + 3A 2 B + 3AB 2 + B 3
(g) (A + B)(A − B)=A 2 − B 2
Exercise 2.1.27 Consider the matrices A = ⎣
D =
[ −1 1
4 −3
] [ 1
,E =
3
]
⎡
1 2
3 2
1 −1
⎤
[
⎦,B =
Find the following if possible. If it is not possible explain why.
(a) −3A T
(b) 3B − A T
(c) E T B
(d) EE T
(e) B T B
(f) CA T
(g) D T BE
2 −5 2
−3 2 1
] [ 1 2
,C =
5 0
]
,
Exercise 2.1.28 Let A be an n × n matrix. Show A equals the sum of a symmetric and a skew symmetric
matrix. Hint: Show that 1 2
(
A T + A ) is symmetric and then consider using this as one of the matrices.
Exercise 2.1.29 Show that the main diagonal of every skew symmetric matrix consists of only zeros.
Recall that the main diagonal consists of every entry of the matrix which is of the form a ii .
Exercise 2.1.30 Prove 3. That is, show that for an m × nmatrixA,ann× p matrix B, and scalars r,s, the
following holds:
(rA+ sB) T = rA T + sB T
Exercise 2.1.31 Prove that I m A = AwhereAisanm× nmatrix.