Team Maths -
Created by: Team Maths -, Last Updated: May 22, 2024


What are Matrices?

A matrix is a collection of elements (numbers, symbols, or expressions) organized in a grid of rows and columns. Each element in a matrix is identified by its position in the grid, typically denoted as aija_{ij}, where ii is the row number and jj is the column number.

Types of Matrices

1. Row Matrix

A row matrix has only one row.

Row Matrix

2. Column Matrix

A column matrix has only one column.

Column Matrix

3. Square Matrix

A square matrix has the same number of rows and columns.

Square Matrix

4. Diagonal Matrix

A diagonal matrix is a square matrix where all elements outside the main diagonal are zero.

Diagonal Matrix

5. Identity Matrix

An identity matrix is a diagonal matrix where all diagonal elements are 1. It is denoted by I.

Identity Matrix

6. Zero Matrix (Null Matrix)

A zero matrix has all its elements equal to zero.


7. Symmetric Matrix

A symmetric matrix is a square matrix that is equal to its transpose (A=AT).

Symmetric Matrix

8. Skew-Symmetric Matrix

A skew-symmetric matrix is a square matrix where the transpose is equal to its negative (AT=−A).

Skew-Symmetric Matrix

9. Upper Triangular Matrix

An upper triangular matrix is a square matrix where all elements below the main diagonal are zero.

Upper Triangular Matrix

10. Lower Triangular Matrix

A lower triangular matrix is a square matrix where all elements above the main diagonal are zero.

Lower Triangular Matrix

11. Orthogonal Matrix

An orthogonal matrix is a square matrix whose rows and columns are orthogonal unit vectors (orthonormal vectors). The transpose of an orthogonal matrix is also its inverse (AT=A−1A^T = A^{-1}AT=A−1).

Orthogonal Matrix

12. Singular Matrix

A singular matrix is a square matrix that does not have an inverse. Its determinant is zero.

13. Non-Singular Matrix

A non-singular matrix is a square matrix that has an inverse. Its determinant is non-zero.

Matrix Operations

Matrix operations are fundamental in linear algebra and are used extensively in various fields such as physics, engineering, computer science, and economics. Here are the key matrix operations:

1. Matrix Addition

To add two matrices A and B, they must have the same dimensions. The sum is obtained by adding corresponding elements.

additionaddition 2

2. Matrix Subtraction

Similar to addition, to subtract matrix B from matrix A, they must have the same dimensions. The difference is obtained by subtracting corresponding elements.


3. Scalar Multiplication

Multiplying a matrix A by a scalar k involves multiplying each element of A by k.

Scalar Multiplication

4. Matrix Multiplication

The product of two matrices A and B is defined if the number of columns in A equals the number of rows in B. The element cijc_{ij}cij​ in the resulting matrix C is the dot product of the i-th row of A and the j-th column of BBB.

Properties of Matrix Multiplication

There are different properties associated with the multiplication of matrices. For any three matrices A, B, and C:

  • AB ≠ BA
  • A(BC) = (AB)C
  • A(B + C) = AB + AC
  • (A + B)C = AC + BC
  • A = A = AIₙ, for identity matrices I𝑚 and Iₙ.
  • Aₘ ₓ ₙOₙ ₓ ₚ=Oₘ ₓ ₚ, where O is a null matrix.

5. Matrix Transposition

The transpose of a matrix A is obtained by swapping its rows with its columns. The transpose of A is denoted by Aᵀ

6. Matrix Inversion

The inverse of a square matrix A is denoted by A⁻¹ and is defined as the matrix that, when multiplied by A, results in the identity matrix. Not all matrices have an inverse; a matrix must be non-singular (its determinant is non-zero) to have an inverse.


7. Determinant

The determinant is a scalar value that is a function of a square matrix. It is denoted as det (A) or ∣A∣ and provides important properties, such as whether a matrix is invertible. For a 2×2 matrix:

8. Trace

The trace of a square matrix A is the sum of its diagonal elements. It is denoted by tr(A)

Matrices Formulas

  • A(adj A) = (adj A) A = | A | Iₙ
  • | adj A | = | A |ⁿ⁻¹
  • adj (adj A) = | A |ⁿ⁻² A
  • | adj (adj A) | = | A |⁽ⁿ⁻¹⁾²
  • adj (AB) = (adj B) (adj A)
  • adj (A) = (adj A),
  • adj (kA) = kⁿ⁻¹ (adj A) , k ∈ R
  • adj(In) = In
  • adj 0 = 0
  • A is symmetric ⇒ (adj A) is also symmetric.
  • A is diagonal ⇒ (adj A) is also diagonal.
  • A is triangular ⇒ adj A is also triangular.
  • A is Singular⇒| adj A | = 0
  • A⁻¹ = (1/|A|) adj A
  • (AB)⁻¹ = B⁻¹A⁻¹

Notation of Matrices

Matrix notation is a systematic way of organizing data or numbers into a rectangular array using rows and columns. Each entry in the matrix is typically represented by a variable with two subscripts indicating its position within the matrix.

Here is examples to illustrate matrix notation:

Notation of Matrices

Example 1: 2×3 Matrix

Consider a matrix A of size 2×3 (2 rows and 3 columns)

  • (Row 1, Column 1)
  • (Row 1, Column 2)
  • (Row 1, Column 3)
  • (Row 2, Column 1)
  • (Row 2, Column 2)
  • (Row 2, Column 3)

Important Notes on Matrices:

  • Cofactor of the matrix A is obtained when the minor Mᵢⱼ of the matrix is multiplied with (-1)ᶦ⁺ʲ
  • Matrices are rectangular-shaped arrays.
  • The inverse of matrices is calculated by using the given formula: A-1 = (1/|A|)(adj A).
  • The inverse of a matrix exists if and only if |A| ≠ 0.

What is the Difference Between Matrix and Matrices?

A “matrix” is a singular term describing a rectangular array of numbers. “Matrices” is the plural form, referring to multiple such arrays.

What Are Matrices Used For?

Matrices are used to solve systems of linear equations, perform geometric transformations, and handle data in fields like economics, engineering, and computer science.

What is the Definition of Matrices?

Matrices are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns, used in various mathematical computations.

What Are the 4 Types of Matrices?

The four types of matrices include square, diagonal, scalar, and identity matrices, each having unique properties and applications.

How Matrix is Used in Real Life?

In real life, matrices are used for graphics transformations, cryptography, economic modeling, and network analysis, simplifying complex calculations.

What Are the 7 Types of Matrix?

The seven types of matrices are square, rectangular, diagonal, scalar, identity, zero, and triangular matrices, each serving specific mathematical purposes.

Is Matrices Algebra or Calculus?

Matrices belong to the field of algebra, specifically linear algebra, which deals with vectors, vector spaces, and linear transformations.

Is Matrix Calculus or Algebra?

Matrix operations are primarily algebraic. Matrix calculus refers to applying calculus operations like differentiation to matrices.

Why Are Matrices Important in Real Life?

Matrices are crucial in real life for modeling physical systems, performing data analysis, and optimizing processes across various disciplines.

How to Understand Matrices?

To understand matrices, start with basic operations like addition and multiplication, then explore their applications in solving linear equations and transformations.

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