M2 Chapter: Determinants - Your Ultimate Study Guide!
Hey everyone! Welcome to the study notes for Determinants. You might be wondering, "What on earth is a determinant?" Don't worry, it's not as scary as it sounds!
Think of a determinant as a special secret number that we can calculate from any square matrix (like a 2x2 or 3x3 grid of numbers). This single number is incredibly powerful. It can tell us if a system of linear equations has a unique solution, help us find the area of a triangle, and much more. In this chapter, we'll unlock the secrets of how to find this number and use its power!
What is a Determinant? The Basics
A determinant is a scalar value (just a single number) that is calculated from the elements of a square matrix.
Key Points:
- Determinants only exist for square matrices (2x2, 3x3, etc.). You can't find the determinant of a non-square matrix (like a 2x3).
- The notation for the determinant of a matrix A can be written in two ways: det(A) or |A|.
Common Mistake Alert!
When you see |A|, it means "the determinant of matrix A". It does NOT mean the absolute value of the matrix. This is a very common point of confusion, so be careful!
Determinants of Order 2 (2x2 Matrices)
This is the foundation for everything else, so let's get it right! It's super simple.
For a general 2x2 matrix A:
$$ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$The determinant is calculated by multiplying the elements on the main diagonal (top-left to bottom-right) and subtracting the product of the elements on the other diagonal (top-right to bottom-left).
The Formula:
$$ \text{det}(A) = |A| = ad - bc $$Memory Aid: The "Downhill - Uphill" Rule
Imagine you're skiing on the matrix.
- Multiply the numbers on the "downhill" slope (↘): $$ a \times d $$
- Multiply the numbers on the "uphill" slope (↗): $$ b \times c $$
- Calculate: Downhill - Uphill
Step-by-Step Example:
Find the determinant of the matrix B:
$$ B = \begin{pmatrix} 4 & 2 \\ 1 & 3 \end{pmatrix} $$- Identify a, b, c, d: Here, a=4, b=2, c=1, d=3.
- Downhill product (ad): $$ 4 \times 3 = 12 $$
- Uphill product (bc): $$ 2 \times 1 = 2 $$
- Subtract: $$ \text{det}(B) = 12 - 2 = 10 $$
So, the determinant is 10. See? Not so bad!
Key Takeaway
For a 2x2 matrix $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$, the determinant is always ad - bc.
Determinants of Order 3 (3x3 Matrices)
Now we're moving up to 3x3 matrices. The process is a bit longer, but it's just a few more steps. Don't worry if this seems tricky at first; we'll break it down into a clear, repeatable process. There are two common methods.
Method 1: Expansion by Minors (The Formal Method)
This method works for any size of square matrix and is the most reliable one to learn. The idea is to break the 3x3 problem down into smaller 2x2 problems.
The Sign Checkerboard
First, you need to know the "checkerboard" of signs for a 3x3 matrix. It always starts with a `+` in the top-left corner.
$$ \begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \end{pmatrix} $$We'll use these signs in our calculation. For simplicity, let's always expand along the first row. So, we'll be using the signs: +, -, +.
Step-by-Step Process:
Let's find the determinant of matrix C:
$$ C = \begin{pmatrix} 1 & 5 & 3 \\ 2 & 4 & 7 \\ 4 & 6 & 2 \end{pmatrix} $$We will expand along the first row (elements 1, 5, and 3).
-
First Element (1):
- The sign from the checkerboard is +.
- Cover up the row and column containing the 1. $$ \begin{pmatrix} \Box & \Box & \Box \\ \Box & 4 & 7 \\ \Box & 6 & 2 \end{pmatrix} $$
- Find the determinant of the remaining 2x2 matrix (this is called the "minor"): $$ | \begin{smallmatrix} 4 & 7 \\ 6 & 2 \end{smallmatrix} | = (4)(2) - (7)(6) = 8 - 42 = -34 $$
- The first part of our answer is: $$ \mathbf{(+1)} \times (-34) = -34 $$
-
Second Element (5):
- The sign from the checkerboard is -.
- Cover up the row and column containing the 5. $$ \begin{pmatrix} \Box & \Box & \Box \\ 2 & \Box & 7 \\ 4 & \Box & 2 \end{pmatrix} $$
- Find the determinant of the minor: $$ | \begin{smallmatrix} 2 & 7 \\ 4 & 2 \end{smallmatrix} | = (2)(2) - (7)(4) = 4 - 28 = -24 $$
- The second part of our answer is: $$ \mathbf{(-5)} \times (-24) = 120 $$
-
Third Element (3):
- The sign from the checkerboard is +.
- Cover up the row and column containing the 3. $$ \begin{pmatrix} \Box & \Box & \Box \\ 2 & 4 & \Box \\ 4 & 6 & \Box \end{pmatrix} $$
- Find the determinant of the minor: $$ | \begin{smallmatrix} 2 & 4 \\ 4 & 6 \end{smallmatrix} | = (2)(6) - (4)(4) = 12 - 16 = -4 $$
- The third part of our answer is: $$ \mathbf{(+3)} \times (-4) = -12 $$
-
Combine the results:
Now, just add up the three parts.
$$ \text{det}(C) = -34 + 120 - 12 = 74 $$
The determinant is 74. The key is to be slow and careful with your arithmetic!
Method 2: Sarrus's Rule (The 3x3 Shortcut)
This is a visual shortcut that only works for 3x3 matrices. Many students find it easier and faster.
Step-by-Step Process:
Let's use the same matrix C:
$$ C = \begin{pmatrix} 1 & 5 & 3 \\ 2 & 4 & 7 \\ 4 & 6 & 2 \end{pmatrix} $$- Copy the first two columns and write them to the right of the matrix. $$ \begin{pmatrix} 1 & 5 & 3 \\ 2 & 4 & 7 \\ 4 & 6 & 2 \end{pmatrix} \begin{matrix} 1 & 5 \\ 2 & 4 \\ 4 & 6 \end{matrix} $$
- Multiply along the three "downhill" diagonals (↘) and add them together.
- Green: $$ (1 \times 4 \times 2) = 8 $$
- Blue: $$ (5 \times 7 \times 4) = 140 $$
- Red: $$ (3 \times 2 \times 6) = 36 $$
- Sum of Downhills: $$ 8 + 140 + 36 = 184 $$
- Multiply along the three "uphill" diagonals (↗) and add them together.
- Green: $$ (4 \times 4 \times 3) = 48 $$
- Blue: $$ (6 \times 7 \times 1) = 42 $$
- Red: $$ (2 \times 2 \times 5) = 20 $$
- Sum of Uphills: $$ 48 + 42 + 20 = 110 $$
- Subtract the sums: (Sum of Downhills) - (Sum of Uphills) $$ \text{det}(C) = 184 - 110 = 74 $$
We get the same answer, 74!
Key Takeaway
For 3x3 matrices, you have two choices:
- Expansion by Minors: The formal method. It's systematic and always works.
- Sarrus's Rule: A quick visual shortcut, but remember it's only for 3x3.
Practice both and see which one you prefer!
Properties and Applications of Determinants
Okay, so we can calculate this number. But what is it for? Determinants have some very important properties and uses, especially for solving systems of equations.
Key Properties
-
Determinant of a Product:
The determinant of the product of two matrices is the product of their determinants. This is a super useful property!
$$ \mathbf{|AB| = |A||B|} $$ -
Singular and Non-Singular Matrices:
This is the most important concept to understand for this chapter.
- If det(A) = 0, the matrix A is called a singular matrix. This tells you that the system of equations represented by the matrix does not have a unique solution.
- If det(A) ≠ 0, the matrix A is called a non-singular matrix. This means a unique solution exists.
Did you know?
Geometrically, the absolute value of the determinant of a 2x2 matrix gives the area of the parallelogram formed by its column vectors. For a 3x3 matrix, it gives the volume of the parallelepiped! It tells you how much a transformation scales area or volume.
Application: Cramer's Rule
Cramer's Rule is a formula that uses determinants to solve a system of linear equations directly. It looks complicated, but it's just a recipe to follow. It only works if the system has a unique solution, which means the determinant of the coefficient matrix must be non-zero!
Cramer's Rule for a 2x2 System
Consider the system:
$$ a_1x + b_1y = c_1 $$ $$ a_2x + b_2y = c_2 $$- Find D, the determinant of the coefficient matrix. $$ D = \left| \begin{matrix} a_1 & b_1 \\ a_2 & b_2 \end{matrix} \right| $$
- Find Dx. Replace the x-column (the 'a's) with the constants (the 'c's). $$ D_x = \left| \begin{matrix} c_1 & b_1 \\ c_2 & b_2 \end{matrix} \right| $$
- Find Dy. Replace the y-column (the 'b's) with the constants. $$ D_y = \left| \begin{matrix} a_1 & c_1 \\ a_2 & c_2 \end{matrix} \right| $$
- Solve for x and y. $$ \mathbf{x = \frac{D_x}{D}} \quad \text{and} \quad \mathbf{y = \frac{D_y}{D}} \quad (\text{provided } D \neq 0) $$
Cramer's Rule for a 3x3 System
It's the exact same idea, just with more calculations. For a system with variables x, y, and z:
- D is the determinant of the 3x3 coefficient matrix.
- Dx is the determinant of the matrix where the x-column is replaced by the constants.
- Dy is the determinant of the matrix where the y-column is replaced by the constants.
- Dz is the determinant of the matrix where the z-column is replaced by the constants.
- The solution is: $$ \mathbf{x = \frac{D_x}{D}}, \quad \mathbf{y = \frac{D_y}{D}}, \quad \mathbf{z = \frac{D_z}{D}} \quad (\text{provided } D \neq 0) $$
Step-by-Step Example (2x2):
Solve the system using Cramer's Rule:
$$ 2x + 3y = 7 $$ $$ x - 4y = -2 $$- Find D: $$ D = \left| \begin{matrix} 2 & 3 \\ 1 & -4 \end{matrix} \right| = (2)(-4) - (3)(1) = -8 - 3 = -11 $$ Since D ≠ 0, a unique solution exists!
- Find Dx: $$ D_x = \left| \begin{matrix} 7 & 3 \\ -2 & -4 \end{matrix} \right| = (7)(-4) - (3)(-2) = -28 - (-6) = -22 $$
- Find Dy: $$ D_y = \left| \begin{matrix} 2 & 7 \\ 1 & -2 \end{matrix} \right| = (2)(-2) - (7)(1) = -4 - 7 = -11 $$
- Solve: $$ x = \frac{D_x}{D} = \frac{-22}{-11} = 2 $$ $$ y = \frac{D_y}{D} = \frac{-11}{-11} = 1 $$
The solution is x=2, y=1. You can plug these back into the original equations to check!
Chapter Summary
You've made it! Let's do a quick recap.
- A determinant is a special number calculated from a square matrix, written as det(A) or |A|.
- For a 2x2 matrix, the determinant is ad - bc.
- For a 3x3 matrix, you can use Expansion by Minors or the Sarrus's Rule shortcut.
- If det(A) = 0, the matrix is singular, and the related system of equations has no unique solution.
- Cramer's Rule uses determinants to provide a direct formula for solving systems of linear equations.
Determinants are a fundamental tool in algebra. Mastering the calculations now will make later topics much easier. Keep practicing, and you'll be a pro in no time! Good luck!