Algebra, Trigonometry, and Geometry Revision Study Guide

Fundamental Definitions and Types of Matrices

Order of the Matrix: If a matrix has $m$ rows and $n$ columns, it is of order $m \times n$ . * Example: Matrix $A = \begin{pmatrix} 2 & 3 & -1 \ 0 & -4 & 5 \end{pmatrix}$ is of order $2 \times 3$ . The element $a_{11} = 2$ , $a_{12} = 3$ , and $a_{23} = 5$ .
The Row Matrix: A matrix consisting of exactly one row and any number of columns. Example: $(1, 2, 3)$ .
The Column Matrix: A matrix consisting of exactly one column and any number of rows. Examples: $\begin{pmatrix} 1 \ 0 \end{pmatrix}$ and $\begin{pmatrix} 3 \ -1 \ 5 \end{pmatrix}$ .
The Square Matrix: A matrix where the number of rows equals the number of columns. Example: $\begin{pmatrix} 3 & 2 \ -1 & 0 \end{pmatrix}$ .
The Diagonal Matrix: A square matrix where all elements are zero except for the main diagonal, where at least one element is not zero. Examples: $\begin{pmatrix} 3 & 0 \ 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} 1 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & -5 \end{pmatrix}$ .
The Zero Matrix ( $O$ ): A matrix in which all elements are zero. Examples: $\begin{pmatrix} 0 & 0 & 0 \end{pmatrix}$ or $\begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix}$ .
The Unit Matrix ( $I$ ): A diagonal matrix where all elements of the main diagonal are equal to one. Examples: $\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix}$ .

Matrix Operations and Transposition

Matrix Transpose: The transpose of matrix $A$ , denoted by $A^t$ , is created by replacing rows with columns and columns with rows. * Example: If $A = \begin{pmatrix} 5 & -2 & 1 \ 3 & 4 & 6 \end{pmatrix}$ , then $A^t = \begin{pmatrix} 5 & 3 \ -2 & 4 \ 1 & 6 \end{pmatrix}$ .
Symmetric Matrix: A square matrix $A$ is symmetric if and only if $A = A^t$ . * Example: $A = \begin{pmatrix} 1 & -2 & 5 \ -2 & 3 & 6 \ 5 & 6 & 4 \end{pmatrix}$ is symmetric because $A^t = A$ .
Skew-Symmetric Matrix: A square matrix $A$ is skew-symmetric if and only if $A = -A^t$ . * Example: $A = \begin{pmatrix} 0 & -3 & 2 \ 3 & 0 & 4 \ -2 & -4 & 0 \end{pmatrix}$ is skew-symmetric because $A^t = \begin{pmatrix} 0 & 3 & -2 \ -3 & 0 & -4 \ 2 & 4 & 0 \end{pmatrix} = -A$ .
The Equality of Two Matrices: Two matrices $A$ and $B$ are equal if they have the same order and their corresponding elements are equal ( $a_{ij} = b_{ij}$ ). * Example: Given $\begin{pmatrix} x & 2 & -3 \ y - x & 1 & -2 \end{pmatrix} = \begin{pmatrix} 3 & 2 & -3 \ 5 & 1 & -2 \end{pmatrix}$ , then $x = 3$ and $y - x = 5$ . Solving for $y$ gives $y = 8$ .
Scalar Multiplication: To multiply a real number by a matrix, multiply that number by every element in the matrix. * Example: If $A = \begin{pmatrix} -4 & 1 & 5 \ 2 & -3 & -1 \end{pmatrix}$ , then $2A = \begin{pmatrix} -8 & 2 & 10 \ 4 & -6 & -2 \end{pmatrix}$ .
Adding and Subtracting Matrices: Matrices must be of the same order $m \times n$ . * $A + B$ : A matrix where every element is the sum of corresponding elements in $A$ and $B$ . * $A - B = A + (-B)$ : A matrix where every element is the sum of corresponding elements in $A$ and $-B$ . * Example: $A = \begin{pmatrix} 3 & -2 & 2 \ -2 & 5 & -4 \end{pmatrix}$ , $B = \begin{pmatrix} 0 & -3 & 4 \ -1 & 2 & -2 \end{pmatrix}$ . $A - B = \begin{pmatrix} 3 & 1 & -2 \ -1 & 3 & -2 \end{pmatrix}$ .
Multiplying Matrices: For matrices $A$ (order $m \times l$ ) and $B$ (order $r \times n$ ), product $AB$ is possible if and only if $l = r$ (columns of $A$ = rows of $B$ ). The resulting matrix is of order $m \times n$ . * Example: $A = \begin{pmatrix} 2 & -1 & 0 \ 3 & 1 & -2 \end{pmatrix}$ ( $2 \times 3$ ) and $B = \begin{pmatrix} 3 & 2 \ -1 & 0 \ 4 & 5 \end{pmatrix}$ ( $3 \times 2$ ). The product $AB$ is order $2 \times 2$ . * Calculation: $AB = \begin{pmatrix} (2)(3)+(-1)(-1)+(0)(4) & (2)(2)+(-1)(0)+(0)(5) \ (3)(3)+(1)(-1)+(-2)(4) & (3)(2)+(1)(0)+(-2)(5) \end{pmatrix} = \begin{pmatrix} 7 & 4 \ 0 & -4 \end{pmatrix}$ .
Matrix Properties Remarks: * $(A + B)^t = A^t + B^t$ * $(AB)^t = B^t A^t$

Determinants and Inverses

$2 \times 2$ Determinants: If $A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix}$ , then $|A| = a_{11}a_{22} - a_{21}a_{12}$ . * Example: $|\begin{pmatrix} 4 & -7 \ 2 & 6 \end{pmatrix}| = (4 \times 6) - (2 \times -7) = 24 + 14 = 38$ .
$3 \times 3$ Determinants: Expand using minor determinants along any row/column following the sign pattern $\begin{pmatrix} + & - & + \ - & + & - \ + & - & + \end{pmatrix}$ . * Formula: $|A| = a_{11} \begin{vmatrix} a_{22} & a_{23} \ a_{32} & a_{33} \end{vmatrix} - a_{12} \begin{vmatrix} a_{21} & a_{23} \ a_{31} & a_{33} \end{vmatrix} + a_{13} \begin{vmatrix} a_{21} & a_{22} \ a_{31} & a_{32} \end{vmatrix}$ . * Triangular Matrix Property: Determinant equals the product of main diagonal elements: |\begin{pmatrix} a_{11} & a_{12} & a_{13} \ 0 & a_{22} & a_{23} \ 0 & 0 & a_{33} \end{vmatrix}| = a_{11}a_{22}a_{33}.
Area of Triangle $XYZ$ : If $X(a, b)$ , $Y(c, d)$ , and $Z(e, f)$ , the area is $|A|$ where $A = \frac{1}{2} \begin{vmatrix} a & b & 1 \ c & d & 1 \ e & f & 1 \end{vmatrix}$ . If $|A| = 0$ , points are collinear.
Multiplicative Inverse ( $A^{-1}$ ): Exists if $|A| = \Delta \neq 0$ . For $2 \times 2$ matrix $A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$ , $A^{-1} = \frac{1}{\Delta} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}$ . * Example: $A = \begin{pmatrix} -2 & 2 \ 3 & -4 \end{pmatrix}$ . $\Delta = (-2 \times -4) - (3 \times 2) = 2$ . $A^{-1} = \frac{1}{2} \begin{pmatrix} -4 & -2 \ -3 & -2 \end{pmatrix} = \begin{pmatrix} -2 & -1 \ -\frac{3}{2} & -1 \end{pmatrix}$ .

Systems of Equations

Simultaneous Equations ( $2 \times 2$ ): Solving $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$ . * Cramer's Rule: $\Delta = \begin{vmatrix} a_1 & b_1 \ a_2 & b_2 \end{vmatrix}$ , $\Delta_x = \begin{vmatrix} c_1 & b_1 \ c_2 & b_2 \end{vmatrix}$ , $\Delta_y = \begin{vmatrix} a_1 & c_1 \ a_2 & c_2 \end{vmatrix}$ . Then $x = \frac{\Delta_x}{\Delta}$ and $y = \frac{\Delta_y}{\Delta}$ . * Matrix Inverse Method: Write as $AX = C$ where $A = \begin{pmatrix} a_1 & b_1 \ a_2 & b_2 \end{pmatrix}$ , $X = \begin{pmatrix} x \ y \end{pmatrix}$ , and $C = \begin{pmatrix} c_1 \ c_2 \end{pmatrix}$ . Then $X = A^{-1}C$ .
Simultaneous Equations ( $3 \times 3$ ): For equations with variables $x, y, z$ , calculate $\Delta$ , $\Delta_x$ , $\Delta_y$ , and $\Delta_z$ using determinants. $x = \frac{\Delta_x}{\Delta}$ , $y = \frac{\Delta_y}{\Delta}$ , $z = \frac{\Delta_z}{\Delta}$ .

Linear Programming and Optimization

Boundary Lines: * Inequalities with $\geq$ or $\leq$ are represented by solid lines (boundary is part of the solution). * Inequalities with > or < are represented by dashed lines (boundary is not part of the solution).
Graphical Solution Set: Determine the region for each individual inequality. The overlap of all regions is the common solution set.
Objective Function Optimization: To maximize or minimize $P = ax + by$ , evaluate the function at the corner points (vertices) of the shaded solution region. * Example: Region defined by $x \geq 0, y \geq 0, x + 2y \leq 8, 3x + 2y \leq 12$ . Objective function $P = 50x + 75y$ . Evaluated at: $A(4, 0) \rightarrow P = 200$ , $B(2, 3) \rightarrow P = 325$ , $C(0, 4) \rightarrow P = 300$ . Maximum value is 325 at $(2, 3)$ .

Trigonometric Measures and Conversion

Radian and Degree Relation: $\frac{\theta_{rad}}{\pi} = \frac{x^\circ}{180^\circ}$ . * Converter: $x^\circ = \theta_{rad} \times \frac{180^\circ}{\pi}$ . * Converter: $\theta_{rad} = x^\circ \times \frac{\pi}{180^\circ}$ .
Arc Length ( $l$ ): If $r$ is radius and $\theta_{rad}$ is the central angle, $l = \theta_{rad}r$ .
Trigonometric Ratios on Unit Circle: For point $(x, y)$ , $\sin(\theta) = y$ , $\cos(\theta) = x$ , $\tan(\theta) = \frac{y}{x}$ .
Relation between $\theta$ and $-\theta$ : * $\sin(-\theta) = -\sin(\theta)$ , $\csc(-\theta) = -\csc(\theta)$ . * $\cos(-\theta) = \cos(\theta)$ , $\sec(-\theta) = \sec(\theta)$ . * $\tan(-\theta) = -\tan(\theta)$ , $\cot(-\theta) = -\cot(\theta)$ .

Trigonometric Identities and Equations

Pythagorean Identities: * $\sin^2(\theta) + \cos^2(\theta) = 1 \Rightarrow \sin^2(\theta) = 1 - \cos^2(\theta)$ , $\cos^2(\theta) = 1 - \sin^2(\theta)$ . * $1 + \tan^2(\theta) = \sec^2(\theta) \Rightarrow \sec^2(\theta) - \tan^2(\theta) = 1$ . * $\cot^2(\theta) + 1 = \csc^2(\theta) \Rightarrow \csc^2(\theta) - \cot^2(\theta) = 1$ .
General Solutions for Trigonometric Equations ( $\beta$ is smallest positive angle, $n \in \mathbb{Z}$ ): * $\cos(\theta) = a \Rightarrow \theta = \pm\beta + 2\pi n$ . * $\sin(\theta) = a \Rightarrow \theta = \beta + 2\pi n$ or $\theta = (\pi - \beta) + 2\pi n$ . * $\tan(\theta) = a \Rightarrow \theta = \beta + \pi n$ .
Quadrantal Angle General Solutions: * $\sin(\theta) = 0 \Rightarrow \theta = \pi n$ * $\sin(\theta) = 1 \Rightarrow \theta = \frac{\pi}{2} + 2\pi n$ * $\cos(\theta) = 0 \Rightarrow \theta = \frac{\pi}{2} + \pi n$ * $\cos(\theta) = 1 \Rightarrow \theta = 2\pi n$
Angles in Quadrants: * 1st Quadrant: $\theta = \beta$ * 2nd Quadrant: $\theta = 180^\circ - \beta$ * 3rd Quadrant: $\theta = 180^\circ + \beta$ * 4th Quadrant: $\theta = 360^\circ - \beta$

Geometric Applications of Trigonometry

Solving Right-Angled Triangles: Use Pythagoras ( $AC^2 = AB^2 + BC^2$ ) and ratios (SOH CAH TOA).
Angles of Elevation and Depression: * Elevation: Point of target is above horizontal line. * Depression: Point of target is below horizontal line. * Elevation angle of $B$ from $A$ equals depression angle of $A$ from $B$ .
Triangles and Polygons Areas: * General Triangle: $\frac{1}{2} \text{base} \times \text{height}$ or $\frac{1}{2} ab \sin(\theta)$ . * Equilateral Triangle: $\frac{\sqrt{3}}{4}s^2$ . * Regular Hexagon: $\frac{3\sqrt{3}}{2}s^2$ . * Quadrilateral: $\frac{1}{2} d_1 d_2 \sin(\theta)$ . * Regular Polygon ( $n$ sides, side $x$ ): Area $A = \frac{1}{4} nx^2 \cot(\frac{\pi}{n})$ .
Circular Sector and Segment: * Sector Area: $\frac{1}{2} lr = \frac{1}{2} r^2 \theta_{rad}$ . * Sector Perimeter: $l + 2r$ . * Segment Area: $\frac{1}{2} r^2 [\theta_{rad} - \sin(\theta)]$ . * Chord Length: $2r \sin(\frac{\theta}{2})$ .

Vector Geometry

Definitions: A directed line segment has starting/ending points and direction ( $\vec{AB} = -\vec{BA}$ ).
Equivalence: $\vec{AB} = \vec{CD}$ if they have the same norm (length) and same direction.
Norm: For point $A(x, y)$ , $\lVert \vec{A} \rVert = \sqrt{x^2 + y^2}$ .
Polar Form: Expressed as $(\lVert \vec{A} \rVert, \theta)$ . * Components: $x = \lVert \vec{A} \rVert \cos(\theta)$ , $y = \lVert \vec{A} \rVert \sin(\theta)$ .
Unit Vector: A vector with norm 1. Fundamental unit vectors: $\mathbf{i} = (1, 0)$ and $\mathbf{j} = (0, 1)$ .
Vector Operations: * Geometric: $\vec{AB} + \vec{BC} = \vec{AC}$ . * Parallelogram rule: $\vec{AB} + \vec{AC} = \vec{AD}$ where $AD$ is the diagonal. * Algebraic: $\vec{A} + \vec{B} = (x_1+x_2, y_1+y_2)$ .
Parallel and Perpendicular Vectors: * Parallel ( $\vec{A} // \vec{B}$ ): Slopes are equal ( $\frac{y_1}{x_1} = \frac{y_2}{x_2}$ ). * Perpendicular ( $\vec{A} \perp \vec{B}$ ): Product of slopes is $-1$ ( $x_1x_2 + y_1y_2 = 0$ ).

Lines and Physical Applications

Resultant Force: $\vec{F} = \vec{F_1} + \vec{F_2} + \dots$
Relative Velocity: Velocity of $B$ relative to $A$ is $\vec{V_{BA}} = \vec{V_B} - \vec{V_A}$ .
Division of Line Segment: Point $C(x, y)$ divides segment between $(x_1, y_1)$ and $(x_2, y_2)$ with ratio $m_2:m_1$ . * Formula: $x = \frac{m_1x_1 + m_2x_2}{m_1 + m_2}$ , $y = \frac{m_1y_1 + m_2y_2}{m_1 + m_2}$ . * Internally: m_2/m_1 > 0. Externally: m_2/m_1 < 0.
Slope of Straight Line: * From points: $m = \frac{y_2-y_1}{x_2-x_1}$ . * From angle: $m = \tan(\theta)$ . * From equation $ax + by + c = 0$ : $m = -\frac{a}{b}$ .
Different Forms of Line Equation: * Vector Form: $\vec{r} = \vec{A} + k\vec{u}$ . * Cartesian Form: $y - y_1 = m(x - x_1)$ . * Intercept Form: $\frac{x}{a} + \frac{y}{b} = 1$ .
Angle between Lines: $\tan(\theta) = |\frac{m_1 - m_2}{1 + m_1m_2}|$ for $\theta \in [0, \frac{\pi}{2}]$ .
Perpendicular Length from Point $(x_1, y_1)$ to line $ax + by + c = 0$ : * $L = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$ .

Questions & Discussion

Q: Given $B = \begin{pmatrix} 2 & 4 \ 3 & 1 \end{pmatrix}$ and $AB = I$ , find matrix $A$ . * A: $A = B^{-1}$ . Construct $\Delta = (2 \times 1) - (3 \times 4) = -10$ . $A = -\frac{1}{10} \begin{pmatrix} 1 & -4 \ -3 & 2 \end{pmatrix} = \begin{pmatrix} -\frac{1}{10} & \frac{2}{5} \ \frac{3}{10} & -\frac{1}{5} \end{pmatrix}$ .
Q: Does $\begin{pmatrix} x & 4 \ 9 & x \end{pmatrix}$ have no inverse? * A: If $\Delta = x^2 - 36 = 0$ , then $x = \pm 6$ .
Q: Determine if points $(3, 5)$ and $(1, 5)$ satisfy $x + y \leq 8$ . * A: $3 + 5 = 8 \leq 8$ (True) and $1 + 5 = 6 \leq 8$ (True).
Q: Find the area of the triangle with vertices $(1, -2)$ , $(3, 2)$ , $(5, 3)$ . * A: Using determinant formula: $\frac{1}{2} |1(2-3) - (-2)(3-5) + 1(9-10)| = \frac{1}{2} |-1 - 4 - 1| = 3$ . Result: 3 square units.
Q: A person observes a plane at 1000m height with elevation angle $40^\circ$ . Find the observer-plane distance. * A: $\sin(40^\circ) = \frac{1000}{x} \Rightarrow x = \frac{1000}{\sin(40^\circ)} \approx 1556 \text{ meters}$ .
Q: Convert vector $\vec{A} = (3, -\sqrt{3})$ to polar form. * A: $\lVert \vec{A} \rVert = \sqrt{9 + 3} = 2\sqrt{3}$ . $\tan(\theta) = -\frac{\sqrt{3}}{3} \Rightarrow \theta = 330^\circ$ . Form: $(2\sqrt{3}, 330^\circ)$ .