Matrix Arithmetic

Part 4: Advanced Properties

In this section we continue to experiment with the matrices A and B defined in Part 1.

1. Compute (AB)⁵ and A⁵B⁵. What do you see? Explain your observations.
2. For the matrix X defined in the worksheet, compute X² and X³. This is another case in which matrix multiplication does not behave like multiplication of real numbers.
   • What is the property of real numbers in this case that does not carry over to matrices?

   • Explain why this example serves as a counterexample to the property you just named.

   The matrix X is an example of a nilpotent matrix. The name means that some power is zero.
3. In this step we will see one more case in which matrix multiplication does not behave as you might think it would. Enter the definitions for P, Q, and R in the worksheet. Compute PQ and PR. When you compare the results, you will see that another multiplicative property of the real numbers does not carry over to matrices. Which property of real numbers is it?
4. A square matrix with ones on the main diagonal and zeros everywhere else is called an identity matrix. A matrix of any size with all zero entries is called a zero matrix. Some examples of each type are defined in the worksheet. Compute each of the following products:
• AI₄
• I₄A
• B I₄
• AZ₁
• Z₁A
• AZ₂
5. From these multiplications we can see that identity matrices and zero matrices have properties like certain real numbers. Complete the following sentences:
• The identity matrix behaves multiplicatively much like the real number ...
• A zero matrix behaves multiplicatively much like the real number ...
6. Try computing the matrix product AZ₃: What goes wrong? Why?

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