r1 - Some questions possibly relevant to MATA33 B....

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Unformatted text preview: Some questions possibly relevant to MATA33 B. Determine whether each of the following statements is (a) always true, (b) always false, or (c) sometimes true and sometimes false. If always true, explain. If not always true, provide a counterex- ample. The statements are phrased in If ..., then ... form, but you can imagine them being worded differ- ently. For example, the first statement is equivalent to Every matrix can be reduced. 1. If is any matrix, then has a reduced form. Hint: p. 252. 2. If A is lower triangular, then A is invertible. 3. If A is invertible, then A is square. 4. If A has a square (i.e. A 2 exists), then A is square (i.e. A is n n ). 5. If A is square, then A n exists for any natural number n . 6. If D is diagonal, then D is invertible. 7. If A is square and has a zero-row, then A is not invertible. 8. If A is not invertible, then the reduced form of A has at least one zero-row. 9. If the reduced forms of A and B are both R , then A = B . 10. If A is invertible, then ( A- 1 )- 1 = A . Hint: Interpret A- 1 A = I carefully....
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This note was uploaded on 08/14/2011 for the course MAT a33 taught by Professor G during the Spring '11 term at University of Toronto.

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r1 - Some questions possibly relevant to MATA33 B....

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