Determinants | Introduction & Basics

1

What is a Determinant?

A determinant is a special number associated with every square matrix. It is a scalar value computed from the elements of a matrix and encodes important properties about the matrix — such as whether the matrix has an inverse.

Determinant of a matrix A is written as det(A) or |A|.

📘 Definition

For every square matrix A, a unique real number is associated called the determinant of A, denoted by det(A) or |A|. Only square matrices have determinants.

🔢

Scalar Output

A determinant always produces a single number (scalar), not a matrix.

⬛

Square Matrices Only

Determinants are defined only for square matrices (n × n), not rectangular ones.

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Key Property

If |A| ≠ 0, the matrix is called non-singular and has an inverse.

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Determinant of 1×1 and 2×2 Matrices

Order 1×1: For a 1×1 matrix A = [a], the determinant is simply the element itself.

1×1 Determinant

|A| = |[a]| = a

Order 2×2: For a 2×2 matrix, the determinant is the difference of products of diagonals.

2×2 Determinant Formula

a b c d

=

ad − bc

✏️ Solved Example

Find the determinant of A = [[3, 4], [5, 2]]

1

Identify elements: a = 3, b = 4, c = 5, d = 2

2

Apply formula: |A| = ad − bc = (3×2) − (4×5)

3

|A| = 6 − 20 = −14

✏️ Solved Example 2

If |A| = 0, find x for A = [[x, 2], [3, 6]]

1

|A| = 6x − 6 = 0

2

6x = 6 → x = 1

3

Determinant of a 3×3 Matrix

For a 3×3 matrix, we use expansion along the first row (also called Laplace's expansion).

3×3 Matrix — Row Expansion

|A| = a₁₁(a₂₂a₃₃ − a₂₃a₃₂)
− a₁₂(a₂₁a₃₃ − a₂₃a₃₁)
+ a₁₃(a₂₁a₃₂ − a₂₂a₃₁)

💡

Signs follow the pattern +, −, + when expanding along the first row. This alternating sign pattern is crucial — don't miss it!

✏️ Solved Example

Find |A| for A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]

1

Expand along Row 1: |A| = 1·|5,6;8,9| − 2·|4,6;7,9| + 3·|4,5;7,8|

2

= 1·(45−48) − 2·(36−42) + 3·(32−35)

3

= 1·(−3) − 2·(−6) + 3·(−3)

4

= −3 + 12 − 9 = 0

⚠️

When |A| = 0, the matrix is singular (non-invertible). The rows (or columns) are linearly dependent.

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Singular and Non-Singular Matrices

Property	Singular Matrix	Non-Singular Matrix
Determinant	\|A\| = 0	\|A\| ≠ 0
Inverse	Does NOT exist	Exists (A⁻¹)
Rows/Columns	Linearly dependent	Linearly independent
System of equations	No unique solution	Unique solution exists

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Important Results to Remember

📌 Key Formulae

• For scalar k and matrix A of order n: |kA| = kⁿ|A|
• |AB| = |A|·|B| (product rule)
• |Aᵀ| = |A| (determinant of transpose equals determinant)
• If any row or column is zero → |A| = 0
• If any two rows/columns are identical → |A| = 0

Introduction &Basic Determinants

What is a Determinant?

Scalar Output

Square Matrices Only

Key Property

Determinant of 1×1 and 2×2 Matrices

Determinant of a 3×3 Matrix

Singular and Non-Singular Matrices

Important Results to Remember

Introduction &
Basic Determinants