Page 119 - Matrix Analysis & Applied Linear Algebra

3.6 Properties of Matrix Multiplication 113

3.6.2. For all matrices A n×k and B k×n , show that the block matrix

I − BA B
L =
2A − ABA AB − I
2
has the property L = I. Matrices with this property are said to be
involutory, and they occur in the science of cryptography.

3.6.3. For the matrix
1001/31/31/3
 
 0101/31/31/3 
 
 ,
 0011/31/31/3 
A = 
 0001/31/31/3 
0001/31/31/3
 
0001/31/31/3
determine A 300 . Hint: A square matrix C is said to be idempotent
2
when it has the property that C = C. Make use of idempotent sub-
matrices in A.
3.6.4. For every matrix A m×n , demonstrate that the products A A and
∗
AA are hermitian matrices.
∗
3.6.5. If A and B are symmetric matrices that commute, prove that the
product AB is also symmetric. If AB
= BA, is AB necessarily sym-
metric?

3.6.6. Prove that the right-hand distributive property is true.

3.6.7. For each matrix A n×n , explain why it is impossible to find a solution
for X n×n in the matrix equation

AX − XA = I.
Hint: Consider the trace function.

3.6.8. Let y T be a row of unknowns, and let A m×n and b T be known
1×m 1×n
matrices.
T
(a) Explain why the matrix equation y A = b T represents a sys-
tem of n linear equations in m unknowns.
T
(b) How are the solutions for y T in y A = b T related to the
T
solutions for x in A x = b?