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Inverses and Elementary Matrices

#### Part 1: Basic Properties

Enter the matrices **A** and **B** defined in your worksheet.
- Compute the product
**AB**. What is
the relationship between the matrices **A** and **B**?
- In general, if
**A** and **B** are matrices such that
**AB = I**, then **B** is called a *right inverse* for **A**. Similarly, if **BA
= I**, then **B** is a *left inverse *for **A**. If **A** and **B** are square matrices
such that **AB = I** and **BA = I**, then **A** and **B** are *inverses* for each
other. A square matrix that has an inverse is called *invertible*.
In this terminology, what can you say about the matrices **A** and **B** in Step
1 and the relationship between them?
- Enter the matrices
**P** and **Q** defined in your worksheet,
and compute both **PQ** and **QP**. What is the relationship between **P** and **Q**? Must
a left inverse also be a right inverse?
- Use matrix algebra to establish the following
fact:

*If a square matrix ***A** is invertible and **AC = I**, then **C** is the inverse
of **A**.

Why don't the matrices **P** and **Q** in Step 3 contradict this statement?
- There are many ways to determine whether a square
matrix is invertible. One way is the following:

*A square matrix is invertible if and only if its reduced row echelon
form is the identity matrix I.*

Enter the matrices **R** and **S** defined in your worksheet, and use this
criterion to decide whether each is invertible.
(A matrix which is not invertible is called *singular*.
Sometimes a singular matrix is called *noninvertible*
and an invertible matrix is called *nonsingular*.)
- In Step 5 you found that the matrix
**S** is invertible.
Use the inverse command to find its inverse, and call this matrix **T**. Check
that **T** really is the inverse by computing **ST** and **TS**.
- Compute

**(AS)**^{-1},

**A**^{-1}S^{-1}, and

**S**^{-1}A^{-1}.

What do you deduce?
- Compute

**(S**^{T})^{-1} and

**(S**^{-1})^{T}.

What do you deduce?

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