**Linear Algebra (continued)**

**1 Inverses of matrix**

Let A be an nn (square matrix). The inverses of A, denoted by , is an n n matrix such that

If A has an inverse, then A is called a *non-singular matrix* and said to be *invertible*.

If A has no inverse, then A is called a *singular matrix*.

**1.1 Existence and Uniqueness of an inverse**

If the inverse exists, then the inverse is unique.

The inverse of an nn matrix A exists if and only if det(A)0, and

In particular, the inverse of 22 matrix A is

**Example 1.1.** Find the inverse of the matrices

**Solution.**

Since

**Exercise 1.1.** Find the inverse of

**1.2 Using Gauss-Jordan Elimination to find inverses**

The formula (1) is reasonable for inverting 33 matrices by hand. For large matrices, a computational procedure based on Gauss-Jordan elimination is more ecient. This procedure proceeds as follows. To nd the inverse of an invertible matrix A, we must find

We want to reduce A to the identity matrix by row operations and simultaneously apply these operations to I to produce . To accomplish that we shall adjoin the identity matrix to the right side of A, thereby producing a matrix of the form

Then we shall apply row operations to this matrix until the left side is reduced to I. These operations will convert the right side to , so that the nal matrix will have the form

This procedure works because the elementary row operations can be represented by a martix transformation, such that

then

where

**Example 1.2. **Find the inverses of

**Solution.**

Following the procedure, the computations are as follow

Thus,

**Exercise 1.2.** Find the inverse of A in exercise 1.1 by Gauss-Jordon elimination.

**2 General properties of inverses**

**2.1 Inverses of special matrices**

**2.2 Further properties of inverses**

Suppose that A and B are invertible nn matrices and k is any scalar, then

**3 Solving system of linear equations by matrix inversion**

If A is an invertible nn matrix, then the system of equations Ax = b has exactly one solution, namely, .

**Example 3**. Consider the system of linear equations

In matrix form this system can be written as Ax = b, where

A is invertible and

Then the solution of the system is

or

**Exercise 3.** Solve

**4 Powers of a matrix**

If A is a square matrix, then we dene the nonnegative integer powers of A to be

If A is invertible, then we dene the negative integer powers to be

Because this denition parallels that for real numbers, the usual laws of exponents hold, i.e. for any integers r and s

**Example 4.** Let

Then

**Exercise 4. **Consider a matrix

Find and .

**5 Matrix polynomial**

If A is an nn matrix, and if

is any polynomial. By substituting A for x, then we have a matrix polynomial in A as

where I is an nn identity matrix.

**Example 5. **If and

then

**Exercise 5.** Consider and A in exercise 4, Find p(A).

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