# Math 0250 Homework 3

• 0.8 Number 4
The equation AX=0 has the non-trivial solution

so the coefficient matrix A is singular.
• 0.8 Number 5
(A-1BA)2 = (A-1BA)(A-1BA) = A-1B(AA-1)BA = A-1BIBA = A-1BBA = A-1B2A.

In general, (A-1BA)n = A-1BnA.

• 0.8 Number 7

We'll show that uuT is singular by showing that the system (uuT)X has a non-trivial solution. Let's consider two cases. First, if u1=0, then

is a solution, since

On the other hand, if , then

is a non-trivial solution to (uuT)X=0, since

In either case (u1=0 or ), we've exhibited a non-trivial solution, so the coefficient matrix uuT is singular.
• 0.9 Number 1

so the matrix is invertible, and its inverse is
• 0.9 Number 2
Since the first row is twice the second, the matrix is not invertible.
• 0.9 Number 7

Since the left block of the reduced matrix has a row of zeroes, the given matrix is singular.
• 0.9 Number 8

so the the matrix is invertible, and its inverse is

• 0.10.1 Number 3

• 0.10.1 Number 7

• 0.10.2 Number 3
Subtracting twice the last row from the first gives

A cofactor expansion along the last column gives

Expanding in cofactors along the last row gives

• 0.11 Number 1
a
Try to solve the system

The coefficient matrix row reduces to

which is the coefficient matrix of an equivalent homogeneous system, which can be easily solved for x1 and x2 in terms of x3:

Taking x3=1 gives the solution

and the dependency relation

b
The vectors are linearly independent since

c
The matrix

row reduces to the identity matrix, so the columns are independent.

d
The matrix

row reduces to

As in part a, the vectors are linearly dependent, with dependence relation

e
The matrix

row reduces to

so the system AX=0 has the solution

giving the dependence relation

f
The matrix

row reduces to

giving the dependence relation

g
The matrix with the given columns row reduces to a identity matrix, so the columns are linearly independent.

h
The matrix with the given columns row reduces to

which is the coefficient matrix of a system with the non-trivial solution

This gives the dependency relation

Frank Beatrous
1/20/1998