Page 128 - Matrix Analysis & Applied Linear Algebra

122 Chapter 3 Matrix Algebra

3.7.3. For a square matrix A, explain why each of the following statements
must be true.
(a) If A contains a zero row or a zero column, then A is singular.
(b) If A contains two identical rows or two identical columns, then
A is singular.
(c) If one row (or column) is a multiple of another row (or column),
then A must be singular.

3.7.4. Answer each of the following questions.
(a) Under what conditions is a diagonal matrix nonsingular? De-
scribe the structure of the inverse of a diagonal matrix.
(b) Under what conditions is a triangular matrix nonsingular? De-
scribe the structure of the inverse of a triangular matrix.

3.7.5. If A is nonsingular and symmetric, prove that A −1 is symmetric.

3.7.6. If A is a square matrix such that I − A is nonsingular, prove that

A(I − A) −1 =(I − A) −1 A.

3.7.7. Prove that if A is m × n and B is n × m such that AB = I m and
BA = I n , then m = n.

3.7.8. If A, B, and A + B are each nonsingular, prove that

−1
−1
−1

−1
−1
A(A + B) B = B(A + B) A = A + B .
3.7.9. Let S be a skew-symmetric matrix with real entries.
T
(a) Prove that I − S is nonsingular. Hint: x x =0 =⇒ x = 0.
T
(b) If A =(I + S)(I − S) −1 , show that A −1 = A .
3.7.10. For matrices A r×r , B s×s , and C r×s such that A and B are nonsin-
gular, verify that each of the following is true.
−1 −1
A 0 A 0
(a) = −1
0 B 0 B
−1 −1 −1 −1
AC A −A CB
(b) = −1
0 B 0 B