By Frenkel D., Portugal R.

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**Sample text**

7. Show that ( A2 )−1 = ( A−1 )2 . Special types of square matrices 45 ⎡1 4 2⎤ ⎡ −1 0 0 ⎤ ⎡ −1 9 1 ⎤ For 8–12, let C = ⎢⎢ −2 5 0 ⎥⎥ , D = ⎢⎢ 0 5 0 ⎥⎥ , and E = ⎢⎢ 0 5 3 ⎥⎥ . Give answers in fractional form as needed. ⎢⎣ 0 0 −4 ⎥⎦ ⎢⎣ 0 0 −4 ⎥⎦ ⎣⎢ 3 −6 −1⎥⎦ 8. Find C −1. 9. Find D −1. 10. Find E −1 . 11. Show that (C + D )−1 ≠ C −1 + D −1. 12. Show that (CDE )−1 = E −1D −1C −1 . For 13–15, use the calculator’s or singular. MATRX det( command to determine whether the given matrix is nonsingular ⎡3 2 ⎤ 13.

Note: See Chapter 1 for an additional discussion of Gauss-Jordan elimination. EXERCISE 3·6 Use Gauss-Jordon elimination to solve the given system. 1. 2x + 4y = 2 3x − 2 y =1 4 x + 2 y − 6z = 2 2. 3 x − y − 4 z = 7 5x + 2 y − 6z = 5 x − 2y = 1 3. 3 x − 6 y = 3 4 x + 2 y − 6z = 2 4. 3 x − y − 4 z = 7 2 x + y − 3z = 2 2 x + 5y − 8z = 4 5. 2 x + 4 y − 6 z = 2 3 x + 8 y − 13 z = 7 6. x + 2y = 0 2 x − 3y = 0 4 x + 2 y − 6z = 0 7. 3 x − y − 4 z = 0 5x + 2 y − 6z = 0 36 practice makes perfect Linear Algebra 1), the system 4 x + 2 y − 6z = 0 8.

3. The only nonsingular n × n idempotent matrix is the n × n ____________ matrix. 4. A square matrix A is nilpotent if there is an integer p such that AP equals the ____________ matrix. 5. All nilpotent matrices are ____________ (nonsingular, singular). 52 practice makes perfect Linear Algebra For 6–10, respond as indicated. 2⎤ ⎡ 5 6. Show that A = ⎢ ⎥ is involutory. ⎣ −12 −5 ⎦ ⎡ 4 −1 −4 ⎤ ⎢ ⎥ 7. Show that B = ⎢ 3 0 −4 ⎥ is involutory. ⎢⎣ 3 −1 −3 ⎥⎦ 1⎤ ⎡3 8. Show that C = ⎢ is idempotent. 5 2 ⎤ ⎢ ⎥ 9.

### Algebraic methods to compute Mathieu functions by Frenkel D., Portugal R.

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