Complex Numbers and Matrices

• Simplify powers of i using the cycle i, −1, −i, 1
• Add, subtract, and multiply complex numbers in a + bi form
• Add, scalar multiply, and multiply 2x2 matrices, and find a determinant (ad − bc)
• Add and subtract vectors component by component and multiply a vector by a scalar
• Find the magnitude of a vector with the Pythagorean theorem

Complex numbers: the imaginary unit i is defined by i² = −1 (so the square root of −1 is i). Its powers cycle in a period of four: i¹ = i, i² = −1, i³ = −i, i⁴ = 1, and then the pattern repeats, so to simplify a high power of i, divide the exponent by 4 and use the remainder. A complex number is written a + bi, where a is the real part and b is the imaginary part. Add and subtract by combining like parts: (3 + 2i) + (1 − 5i) = 4 − 3i. Multiply using foil and then replace i² with −1: (2 + 3i)(1 − i) = 2 − 2i + 3i − 3i² = 2 + i + 3 = 5 + i. This i² = −1 rule is also why a negative discriminant produces complex (not real) solutions. The square root of a negative number is i times the square root of its absolute value: √(−36) = 6i, so √16 + √(−36) = 4 + 6i (a real part and an imaginary part never combine into one number). Square a complex number with (a + bi)² = a² + 2abi + b²i²: (2 + 3i)² = 4 + 12i − 9 = −5 + 12i, not 4 + 9i² = −5, which drops the middle term. For 'which statement cannot be true' questions about a product, test each statement with simple numbers: i · 3i = −3 shows two imaginary numbers can multiply to a negative real, but a nonzero real number times a non-real number always keeps an imaginary part, and a number times its own conjugate is never negative. Matrices: a matrix is a rectangular array of numbers with rows and columns; a 2x2 matrix has two of each. Add or subtract two matrices of the same dimensions entry by entry. Scalar multiplication multiplies every entry by the same number, zeros and negatives included: −3 times rows [2, −1] and [0, 4] is rows [−6, 3] and [0, −12]. Two matrices are equal only when every pair of entries in the same position is equal, so a matrix equation with unknowns is solved entry by entry: if 3 times rows [x, 1] and [2, y], plus rows [2, 0] and [1, −1], equals rows [14, 3] and [7, 5], then 3x + 2 = 14 and 3y − 1 = 5, so x = 4 and y = 2. To multiply two matrices, the number of columns of the first must equal the number of rows of the second, and each entry of the product is a row-times-column dot product; for two 2x2 matrices [[a, b], [c, d]] and [[e, f], [g, h]] the product is [[ae + bg, af + bh], [ce + dg, cf + dh]]. Matrix multiplication is not commutative, so order matters. The determinant of [[a, b], [c, d]] is ad − bc. Vectors: a vector has a horizontal and a vertical component and is written <a, b> or ai + bj. Add and subtract component by component: <2, 5> + <−1, 3> = <1, 8>. A scalar multiplies every component: 3<2, −1> = <6, −3>. The magnitude (length) of <a, b> is √(a² + b²), which is the Pythagorean theorem. For the magnitude of a combination such as 2u + v, find the combined vector first, then its length: with u = <2, 1> and v = <1, −3>, 2u + v = <5, −1> and its magnitude is √26. Top-score extensions: to divide complex numbers, multiply the top and bottom by the conjugate of the denominator (the conjugate of a + bi is a − bi, and (a + bi)(a − bi) = a² + b², a real number). A matrix with m rows and n columns times one with n rows and p columns gives a matrix with m rows and p columns; if the inner numbers differ, the product is not defined. A 2x2 matrix has an inverse only when its determinant is not 0. A system such as 2x + 3y = 8 and x − y = 1 can be written as a matrix equation: the coefficient matrix [[2, 3], [1, −1]] times the column [x, y] equals the column [8, 1]. A vector from point P to point Q has components Q minus P (end minus start).

• For a high power of i, divide the exponent by 4 and keep only the remainder.
• After multiplying complex numbers, always replace i² with −1 before combining terms.
• Before multiplying matrices, check the dimensions: columns of the first must equal rows of the second.
• Vector addition is just two separate additions, one per component.