MathematicsChapter 2: Matrices and Determinants

DETERMINANTS 2x2 AND 3x3

The determinant is a single number attached to a square matrix.

Learning objectives

State the main ideas and notation used in determinants 2x2 and 3x3.
Apply the rules of determinants 2x2 and 3x3 in a clear, step-by-step solution.
Check signs, conditions, and the final answer for mathematical accuracy.

Definition

The determinant is a single number attached to a square matrix. 2x2: det [[a,b],[c,d]] = ad-bc. 3x3: expand along a row or column using cofactors. Cofactor C_{ij} = (-1)^{i+j} times the minor (delete row i, column j, then 2x2 det). Meaning: det tells invertibility (next) and signed volume in the vector chapter. det I = 1. det O = 0.

Mathematical form

C11 = + det[[4,5],[1,1]] = 4-5 = -1

C12 = - det[[3,5],[0,1]] = -(3-0) = -3

det = 1*(-1) + 2*(-3) = -7

I = 1. det O

Where / symbols

DETERMINANTS 2x2 AND 3x3: The determinant is a single number attached to a square matrix. 2x2: det [[a,b],[c,d]] = ad-bc. 3x3: expand along a row or column using cofactors. Cofactor C_{ij} = (-1)^{i+j} times the minor (delete row i, column j, then 2x2 det). Meaning: det tells invertibility (next) and signed volume in the vector chapter. det I = 1. det O = 0.
1. 2x2: ad-bc. Swapping two rows changes sign. If two rows equal, det=0. det(kA)=k^n det A for n x n.
2. 3x3 COFACTORS: Signs + - + on the first row. Pick a row with zeros if you can. Minor = leftover 2x2. Sum a_{ij} C_{ij} along the chosen row.

Overview

Small, then the chessboard signs:

2x2

ad-bc. Swapping two rows changes sign. If two rows equal, det=0. det(kA)=k^n det A for n x n.

Examples [[1,2],[3,4]] -> -2 · Singular if det=0 · det A^T = det A

3x3 COFACTORS

Signs + - + on the first row. Pick a row with zeros if you can. Minor = leftover 2x2. Sum a{ij} C{ij} along the chosen row.

Examples Expand row 1 · Zero entries skip work · Same value any row/col

WHAT — DETERMINANTS 2x2 AND 3x3

RULE — 1. 2x2

ad-bc. Swapping two rows changes sign. If two rows equal, det=0. det(kA)=k^n det A for n x n.

METHOD — 2. 3x3 COFACTORS

Signs + - + on the first row. Pick a row with zeros if you can. Minor = leftover 2x2. Sum a{ij} C{ij} along the chosen row.

WORKED EXAMPLE

Worked: det [[1,2,0],[3,4,5],[0,1,1]]. expand row 1: 1C11 + 2C12 + 0 C11 = + det[[4,5],[1,1]] = 4-5 = -1 C12 = - det[[3,5],[0,1]] = -(3-0) = -3 det = 1(-1) + 2(-3) = -7

How to answer a determinants 2x2 and 3x3 question

Identify the given quantities, notation, and required result.
Choose the matching rule and write it before substitution.
Work one algebraic step at a time, keeping brackets and signs clear.
Check restrictions, simplify the result, and verify it where possible.

Worked example

det [[1,2,0],[3,4,5],[0,1,1]].
expand row 1: 1C11 + 2C12 + 0
C11 = + det[[4,5],[1,1]] = 4-5 = -1
C12 = - det[[3,5],[0,1]] = -(3-0) = -3
det = 1*(-1) + 2*(-3) = -7

In simple words

The sign is (-1)^i+j, not 'always plus'. Chessboard of + and -.

Key points

2x2: ad-bc
3x3: cofactor expansion
ad minus bc
(-1)^i+j

Exam preparation

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