Matrix Operations Formulas

Master linear algebra with comprehensive matrix operation formulas. From basic operations to advanced concepts like eigenvalues and decompositions.

1Matrix Notation and Definitions

General Matrix

Notation:

  • • : Element in row i, column j
  • • : m rows, n columns
  • • : Transpose of A
  • • : Inverse of A

Special Matrices:

  • • Square: m = n
  • • Identity:
  • • Zero:
  • • Diagonal: if

2Basic Matrix Operations

Matrix Addition and Subtraction

Requirement: and must have the same dimensions

Example:

1
2
3
4
+
5
6
7
8
=
6
8
10
12

Scalar Multiplication

Example:

3
1
2
3
4
=
3
6
9
12

3Matrix Multiplication

General Formula

where is and is , resulting in as

Requirements:

  • • Number of columns in A = Number of rows in B
  • • Generally (not commutative)
  • • (associative)

Example (2×2):

1
2
3
4
5
6
7
8
=
1×5+2×7
1×6+2×8
3×5+4×7
3×6+4×8
=
19
22
43
50

4Matrix Transpose

Properties:

  • •
  • •
  • •
  • •

Example:

A =
1
2
3
4
5
6
A^T =
1
4
2
5
3
6

5Determinant

2×2 Matrix Determinant

det(A) =
a
b
c
d
= ad - bc

Example:

3
2
1
4
=3 × 4 - 2 × 1 = 12 - 2 = 10

3×3 Matrix Determinant

Or using cofactor expansion:
where and is the minor

Determinant Properties

  • •
  • •
  • • for n×n matrix
  • •
  • • If any row/column is zero:
  • • Swapping rows changes sign

6Matrix Inverse

Definition and Existence

Exists only if (A is non-singular)

2×2 Matrix Inverse

A-1 = 1/det(A)
d
-b
-c
a
for
A =
a
b
c
d

Example:

A =
4
3
2
1
det(A) = 4 × 1 - 3 × 2 = -2
A-1 = 1/(-2)
1
-3
-2
4
=
-0.5
1.5
1
-2

General Method: Adjugate Formula

where and are cofactors

Properties:

  • •
  • •
  • •

7Eigenvalues and Eigenvectors

Definitions

where is an eigenvalue and is the corresponding eigenvector

Characteristic Equation

Solving this polynomial gives the eigenvalues

For 2×2 Matrix

If
where and

8Advanced Properties and Decompositions

🔢 Matrix Norms

Frobenius:
Spectral:
Max:

📊 Matrix Decompositions

LU:
QR:
SVD:
Eigen:

⚡ Special Matrices

Symmetric:
Orthogonal:
Positive Definite:
Singular:

🧮 Matrix Functions

Trace:
Rank: Number of linearly independent rows
Nullity:

Quick Reference

Common Operations

Addition
Element-wise:
Multiplication
Row × Column:
Transpose
Flip:

Key Properties

Determinant (2×2)
Inverse (2×2)
Eigenvalue
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