Matrix Operations

Part 3: Special Matrices

In this section we explore the properties of certain special matrices: identity, zero, diagonal, and multiplicative inverse matrice. We continue to use the matrices A and B defined in Part 1.

1. 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. An example of each type is defined in the worksheet. Compute each of the following products:
   • A I
   • I A
   • B I
   • A Z
   • Z A
   
   
2. 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 ...
   
   
3. If P and Q are square matrices such that PQ = I and QP = I, then P and Q are called inverse matrices. We write Q = P^-1 and P = Q^-1. A square matrix that has an inverse is called invertible. Use your helper application to determine whether the matrices A and B are invertible and, if so, what their inverse matrices are. [Note: Another name for "invertible" is nonsingular, and a matrix that is not invertible is called singular. Your helper application may use any of these standard names.]

4. Experiment with random square matrices R. Can you find one that is not invertible?

5. A matrix with zeros everywhere except possibly on the main diagonal in called a diagonal matrix. Enter the diagonal matrices D₁ and D₂ defined in your worksheet. Compute D₁⁴. Look at the answer carefully, and explain how you could have computed D₁⁴ in your head.

6. Compute D₁ D₂ and D₂ D₁. What do you conclude about multiplication of diagonal matrices? [Note: This conclusion about diagonal matrices does not hold for arbitrary matrices -- see Part 1.]