Page 39 - A Course in Linear Algebra with Applications

1.2: Operations with Matrices 23

6. Prove the distributive law A(B + C) = AB + AC where A
is m x n, and B and C are n x p.
T T
7. Prove that (Ai?) r = B A where A is m x n and £? is
n x p .

8. Establish the rules c{AB) = (cA)B = A(cB) and (cA) T =
T
cA .
9. If A is an n x n matrix some power of which equals I n,
then A is invertible. Prove or disprove.
10. Show that any two n x n diagonal matrices commute.
11. Prove that a scalar matrix commutes with every square
matrix of the same size.
12. A certain library owns 10,000 books. Each month 20%
of the books in the library are lent out and 80% of the books
lent out are returned, while 10% remain lent out and 10%
are reported lost. Finally, 25% of the books listed as lost the
previous month are found and returned to the library. At
present 9000 books are in the library, 1000 are lent out, and
none are lost. How many books will be in the library, lent
out, and lost after two months ?

T
13. Let A be any square matrix. Prove that \{A + A ) is
T
symmetric, while the matrix \{A — A ) is skew-symmetric.
14. Use the last exercise to show that every square matrix
can be written as the sum of a symmetric matrix and a skew-
symmetric matrix. Illustrate this fact by writing the matrix

(• J -i)

as the sum of a symmetric and a skew-symmetric matrix.
15. Prove that the sum referred to in Exercise 14 is always
unique.