Matrices

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}aij​, where $i$ is the row number and $j$ is the column number.

Types of Matrices

1. Row Matrix

A row matrix has only one row.

2. Column Matrix

A column matrix has only one column.

3. Square Matrix

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

4. Diagonal Matrix

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

5. Identity Matrix

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

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).

8. Skew-Symmetric Matrix

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

9. Upper Triangular Matrix

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

10. Lower Triangular Matrix

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

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).

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.

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.

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3. Scalar Multiplication

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

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${\mathrm{Iₘ}}^{}$ = A = AIₙ, for identity matrices I${}_m$ 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.

AA⁻¹=A⁻¹A=I

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^ᵐ) = (adj A)^ᵐ,
• adj (kA) = k^ⁿ⁻¹ (adj A) , k ∈ R
• adj(I_n) = I_n
• 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:

Example 1: 2×3 Matrix

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

• a₁₁​=1 (Row 1, Column 1)
• $a₁₂=2$ (Row 1, Column 2)
• $a₁₃=3$ (Row 1, Column 3)
• a₂₁​=4 (Row 2, Column 1)
• a₂₂​=5 (Row 2, Column 2)
• a₂₃=6 (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.