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Inverses and Elementary Matrices

Part 3: Singular Matrices

In Part 1 you saw that some square matrices are invertible and some are not. A natural question at this point is "Are there very many invertible matrices, or are they relatively rare?" In particular, do they have to be specially constructed, say, as textbook examples?
  1. You will find in your worksheet commands for constructing a random 5 x 5 matrix and testing it for invertibility. Enter these commands 20 times, and count carefully how many of your 20 random matrices are invertible. What percentage of randomly constructed 5 x 5 matrices would you estimate are invertible?

  2. Can you explain geometrically how your answer could have been predicted in advance? (If you have difficulty imagining 5-dimensional geometry, answer the question for 3 x 3 or 2 x 2 matrices instead.)

  3. The matrix C1, defined in your worksheet, is singular. Verify this statement and explain why the matrix fails to be invertible.

  4. Create your own (non-zero) singular 3 x 3 matrix. Demonstrate that your matrix is singular and explain how you constructed it.

  5. A matrix C2, with an empty column, is defined in your worksheet. Find entries for the missing column which make C2 singular.

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