Page 156 - 35Linear Algebra

156 Matrices

1. Find formulas for the inverses of the following matrices, when they are
not singular:

 
1 a b
(a)  0 1 c 
0 0 1

 
a b c
(b)  0 d e 
0 0 f

When are these matrices singular?

2. Write down all 2×2 bit matrices and decide which of them are singular.
For those which are not singular, pair them with their inverse.

3. Let M be a square matrix. Explain why the following statements are
equivalent:

(a) MX = V has a unique solution for every column vector V .
(b) M is non-singular.

Hint: In general for problems like this, think about the key words:

First, suppose that there is some column vector V such that the equa-
tion MX = V has two distinct solutions. Show that M must be sin-
gular; that is, show that M can have no inverse.

Next, suppose that there is some column vector V such that the equa-
tion MX = V has no solutions. Show that M must be singular.

Finally, suppose that M is non-singular. Show that no matter what
the column vector V is, there is a unique solution to MX = V.

Hint

4. Left and Right Inverses: So far we have only talked about inverses of
square matrices. This problem will explore the notion of a left and
right inverse for a matrix that is not square. Let

0 1 1
A =
1 1 0

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