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Unformatted text preview: MATH 2107 B Complex Numbers Exercises Winter 2008 1 1. Let c1 = 3 + 4i, c2 = 1 - 2i, and c3 = -1 + i. Compute each of the following and simplify as much as possible. (a) c1 + c2 (b) c3 - c1 (c) c1 c2 (d) c2 c3 (e) 4c2 + c2 (f) (-i)c2 (g) 3c1 = ic2 (h) c1 c2 c3 2. Write in the form a + bi. (a) (b) (c) (d)
1+2i 3-4i 12-3i 3-i (2+i)2 i 1 (3+2i)(1+i) 3. Represent each complex number as a point and as a vector in the complex plane. (a) 4 + 2i (b) -3 + i (c) 3 - 2i (d) i(4 + i) 4. Find the modulus of each complex nuimber in Exercise 3. 5. In the complex plane sketch the vecctors corresponding to c and c for c = 2 + 3i and c = -1 + 4i. Geometrically, we can say that C is the reflection of c with respect to the real axis. 6. Let A= 2+i 2i 1 + 2i 2 + 2i -1 + 3i ,C = ,B = -i 0 3-i -2 1-i Compute each of the following and simplify each entry as a + bi. MATH 2107 B (a) A + B (b) (1 - 2i)C (c) AB (d) BC (e) A - 2I2 (f) B (g) AC Complex Numbers Exercises Winter 2008 2 (h) (A + B)C 7. If A = 0 i , compute A2 , A3 and A4 . Give a general rule for An , n a positive integer. i 0 8. Let p(x) denote a polynomial and let A be a square matrix. Then p(A) is called a matrix polynomial or a polynomial in the matrix A. For p(x) = 2x2 + 5x - 3, compute p(A) = 2A2 + 5A - 3In for each of the following (a) A = (b) A = (c) A = (d) A = -3 0 0 -3 1 2 0 1 0 i i 0 1 i 0 0 ...
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