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 scalar value that we calculate from any square matrix (like a \(2 \times 2\) or \(3 \times 3\) grid of numbers). This single number is incredibly powerful. It tells us whether a matrix has an inverse, whether a system of linear equations has a unique solution, and much more. In this chapter, we'll unlock the secrets of how to evaluate determinants and apply their properties!


What is a Determinant? The Basics

A determinant is a scalar value calculated from the elements of a square matrix.

Key Points:

  • Determinants exist only for square matrices (such as \(2 \times 2\), \(3 \times 3\)). You cannot find the determinant of a non-square matrix (like a \(2 \times 3\)).
  • The determinant of a matrix \(A\) is denoted by \(\det(A)\) or \(|A|\).
Common Mistake Alert!

When you see \(|A|\), it represents the determinant of matrix \(A\). It does NOT mean the absolute value of the matrix.


Determinants of Order 2 (\(2 \times 2\) Matrices)

For a general \(2 \times 2\) matrix \(A\):

\(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\)

The determinant is calculated by multiplying the elements along 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:

\(\det(A) = |A| = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc\)

Memory Aid: The "Downhill - Uphill" Rule
  1. Multiply the numbers on the "downhill" diagonal (↘): \(a \times d\)
  2. Multiply the numbers on the "uphill" diagonal (↗): \(b \times c\)
  3. Calculate: Downhill - Uphill

Step-by-Step Example:

Find the determinant of the matrix \(B\):

\(B = \begin{pmatrix} 4 & 2 \\ 1 & 3 \end{pmatrix}\)

  1. Identify \(a, b, c, d\): Here, \(a = 4\), \(b = 2\), \(c = 1\), \(d = 3\).
  2. Downhill product (\(ad\)): \(4 \times 3 = 12\)
  3. Uphill product (\(bc\)): \(2 \times 1 = 2\)
  4. Subtract: \(\det(B) = 12 - 2 = 10\)

Determinants of Order 3 (\(3 \times 3\) Matrices)

For \(3 \times 3\) matrices, there are two primary methods for evaluation: cofactor expansion (expansion by minors) and Sarrus's rule.

Method 1: Expansion by Minors (Cofactor Expansion)

This method breaks a \(3 \times 3\) determinant down into smaller \(2 \times 2\) determinants. It can be performed along any row or column.

The Sign Checkerboard

The sign of the cofactor at row \(i\) and column \(j\) is given by \((-1)^{i+j}\):

\(\begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \end{pmatrix}\)

Step-by-Step Process (Expanding along Row 1):

Let's find the determinant of matrix \(C\):

\(C = \begin{pmatrix} 1 & 5 & 3 \\ 2 & 4 & 7 \\ 4 & 6 & 2 \end{pmatrix}\)

  1. First Element (\(1\)):
    • Sign is \(+\).
    • Minor: \(\begin{vmatrix} 4 & 7 \\ 6 & 2 \end{vmatrix} = (4)(2) - (7)(6) = 8 - 42 = -34\)
    • First term: \((+1) \times (-34) = -34\)

  2. Second Element (\(5\)):
    • Sign is \(-\).
    • Minor: \(\begin{vmatrix} 2 & 7 \\ 4 & 2 \end{vmatrix} = (2)(2) - (7)(4) = 4 - 28 = -24\)
    • Second term: \((-5) \times (-24) = 120\)

  3. Third Element (\(3\)):
    • Sign is \(+\).
    • Minor: \(\begin{vmatrix} 2 & 4 \\ 4 & 6 \end{vmatrix} = (2)(6) - (4)(4) = 12 - 16 = -4\)
    • Third term: \((+3) \times (-4) = -12\)

  4. Combine the results:

    \(\det(C) = -34 + 120 - 12 = 74\)

Method 2: Sarrus's Rule (Shortcut for \(3 \times 3\))

Note: Sarrus's rule works only for \(3 \times 3\) matrices.

  1. Append the first two columns to the right of the matrix:

    \(\begin{pmatrix} 1 & 5 & 3 \\ 2 & 4 & 7 \\ 4 & 6 & 2 \end{pmatrix} \begin{matrix} 1 & 5 \\ 2 & 4 \\ 4 & 6 \end{matrix}\)

  2. Sum of downhill diagonals (↘):

    \((1 \times 4 \times 2) + (5 \times 7 \times 4) + (3 \times 2 \times 6) = 8 + 140 + 36 = 184\)

  3. Sum of uphill diagonals (↗):

    \((4 \times 4 \times 3) + (6 \times 7 \times 1) + (2 \times 2 \times 5) = 48 + 42 + 20 = 110\)

  4. Subtract:

    \(\det(C) = 184 - 110 = 74\)


Properties of Determinants

In the HKDSE M2 syllabus, understanding algebraic and operational properties allows you to simplify determinants efficiently before expanding:

  1. Transpose Property: \(\det(A^T) = \det(A)\)
  2. Multiplication by Scalar: For an \(n \times n\) matrix \(A\) and scalar \(k\), \(\det(kA) = k^n \det(A)\). (For \(3 \times 3\), \(\det(kA) = k^3 \det(A)\)).
  3. Product of Matrices: \(\det(AB) = \det(A)\det(B)\)
  4. Row/Column Operations:
    • Interchanging two rows (or columns) reverses the sign of the determinant: \(\det(A_{\text{swapped}}) = -\det(A)\).
    • Multiplying a single row (or column) by a scalar \(k\) multiplies the determinant by \(k\).
    • Adding a multiple of one row (or column) to another leaves the determinant unchanged: \(R_i \to R_i + kR_j \implies \det(A_{\text{new}}) = \det(A)\).
  5. Zero Determinant Conditions: \(\det(A) = 0\) if:
    • An entire row or column contains only zeros.
    • Two rows or two columns are identical or proportional.
  6. Singular and Non-Singular Matrices:
    • If \(\det(A) = 0\), \(A\) is singular (it has no inverse \(A^{-1}\)).
    • If \(\det(A) \neq 0\), \(A\) is non-singular (it is invertible, and \(\det(A^{-1}) = \frac{1}{\det(A)}\)).

Application: Cramer's Rule

Cramer's Rule is an algebraic method for solving a system of linear equations using determinants. It applies directly when the coefficient matrix has a non-zero determinant (\(D \neq 0\)).

Cramer's Rule for a \(2 \times 2\) System

Consider the system:

\(a_1x + b_1y = c_1\)
\(a_2x + b_2y = c_2\)

  1. Coefficient Determinant \(D\): \(D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}\)
  2. \(D_x\) (replace \(x\)-column with constants): \(D_x = \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}\)
  3. \(D_y\) (replace \(y\)-column with constants): \(D_y = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}\)
  4. Solution: If \(D \neq 0\), \(x = \frac{D_x}{D}\) and \(y = \frac{D_y}{D}\).

Cramer's Rule for a \(3 \times 3\) System

For variables \(x, y, z\), if \(D = \det(\text{coefficient matrix}) \neq 0\):

\(x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}, \quad z = \frac{D_z}{D}\)

Step-by-Step Example (\(2 \times 2\)):

Solve the system using Cramer's Rule:

\(2x + 3y = 7\)
\(x - 4y = -2\)

  1. Calculate \(D\): \(D = \begin{vmatrix} 2 & 3 \\ 1 & -4 \end{vmatrix} = (2)(-4) - (3)(1) = -8 - 3 = -11\)
  2. Calculate \(D_x\): \(D_x = \begin{vmatrix} 7 & 3 \\ -2 & -4 \end{vmatrix} = (7)(-4) - (3)(-2) = -28 - (-6) = -22\)
  3. Calculate \(D_y\): \(D_y = \begin{vmatrix} 2 & 7 \\ 1 & -2 \end{vmatrix} = (2)(-2) - (7)(1) = -4 - 7 = -11\)
  4. Solve:

    \(x = \frac{D_x}{D} = \frac{-22}{-11} = 2\)

    \(y = \frac{D_y}{D} = \frac{-11}{-11} = 1\)

The unique solution is \((x, y) = (2, 1)\).


Chapter Summary

  • A determinant is a scalar value associated with a square matrix, written as \(\det(A)\) or \(|A|\).
  • For a \(2 \times 2\) matrix, \(\det(A) = ad - bc\).
  • For a \(3 \times 3\) matrix, evaluate via Cofactor Expansion or Sarrus's Rule.
  • Key matrix properties include \(\det(AB) = \det(A)\det(B)\), \(\det(A^T) = \det(A)\), and \(\det(kA) = k^n \det(A)\).
  • A matrix is singular if and only if \(\det(A) = 0\).
  • Cramer's Rule provides an explicit determinant-based formula for solving linear systems when \(D \neq 0\).