DETERMINANTS 2x2 AND 3x3

DETERMINANTS 2x2 AND 3x3

What is it?

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.

In simple words

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

Important terms

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.

What you should learn

• 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.

Examples

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

2x2

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

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.

Full school-level explanation

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.

Small, then the chessboard signs:

WHAT — 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.

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

2x2

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

Remember

• ad minus bc

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.

Remember

• (-1)^i+j

Examples

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

How to answer a determinants 2x2 and 3x3 question

How it works

• 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.

Remember

• 2x2: ad-bc
• 3x3: cofactor expansion
• -7
• chessboard signs
• ad minus bc
• (-1)^i+j
• 2x2: ad-bc
• 3x3: cofactor expansion

How the ideas connect

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

Deeper understanding

Go beyond memorising DETERMINANTS 2x2 AND 3x3. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

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

Common mistakes

Exam focus

Core Idea

1. 2x2

2. 3x3 COFACTORS

Worked Example

Exam focus

Quick revision

• 2x2: ad-bc
• 3x3: cofactor expansion
• -7
• chessboard signs
• DETERMINANTS 2x2 AND 3x3: The determinant is a single number attached to a square matrix. 2x2: det [[a,b],[c,d]] = ad-bc. 3x3:
• 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.

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

For long answers on DETERMINANTS 2x2 AND 3x3: define clearly → explain the main parts → give one example → state one exam caution.