2.3soln - Solutions to problems from section 2.3 7 0 4 4 Is...

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Solutions to problems from section 2.3 4. Is the matrix 7 0 4 3 0 1 2 0 9 invertible? Solution: NO. The columns of the matrix above are linearly dependent, for 0 0 0 = 0 7 3 2 + 0 0 0 + 0 4 1 9 . Thus by the inverse matrix theorem (parts (e) and (a)) we know the matrix is singular. 6. Is the matrix 1 5 4 0 3 4 3 6 0 invertible? Solution: NO. Notice 1 5 4 0 3 4 3 6 0 NR 3 = R 3 +3 R 1 −→ 1 5 4 0 3 4 0 9 12 NR 3 = R 3 +3 R 2 −→ 1 5 4 0 3 4 0 0 0 Since the matrix has only 2 pivot positions we know the matrix is singular (by the IMT parts (c) and (a)). 8. Is the matrix 1 3 7 4 0 5 9 6 0 0 2 8 0 0 0 10 invertible? Solution: YES. There are 4 pivot positions, so by the IMT (parts (c) and (a)) we know the matrix above is invertible. 12. Assume all matrices in this problem are n × n matrices. True or false? Justify your answers.
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