Inverses and Elementary Matrices

Part 1: Basic Properties

1. Compute the product AB. What is the relationship between the matrices A and B?
2. 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?
3. 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?
4. 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?
5. 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.)
6. 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.
7. Compute
   (AS)^-1,
   A^-1S^-1, and
   S^-1A^-1.
   What do you deduce?
8. Compute
   (S^T)^-1 and
   (S^-1)^T.
   What do you deduce?

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