Matrices

Matrices – Complete Solutions

3.1 Definition and Order

A matrix is a rectangular arrangement of numbers (or symbols) in horizontal rows and vertical columns, enclosed in brackets. If a matrix has m rows and n columns, it is called a matrix of order m × n.

General Notation:

A = [aᵢⱼ] where i = 1, 2, …, m and j = 1, 2, …, n

Example:

A = [[1, 2, 3], [4, 5, 6]] is a matrix of order 2 × 3

3.2 Types of Matrices

1. Row Matrix:

A matrix with only one row is called a row matrix.

Example: [1 2 3] is a 1 × 3 row matrix

2. Column Matrix:

A matrix with only one column is called a column matrix.

Example: [[1], [2], [3]] is a 3 × 1 column matrix

3. Square Matrix:

A matrix with equal number of rows and columns is called a square matrix.

Example: [[1, 2], [3, 4]] is a 2 × 2 square matrix

4. Diagonal Matrix:

A square matrix where all non-diagonal elements are zero.

Example: [[2, 0, 0], [0, 3, 0], [0, 0, 4]]

5. Identity Matrix (Unit Matrix):

A diagonal matrix where all diagonal elements are 1.

Example: I = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]

6. Null (Zero) Matrix:

A matrix where all elements are zero.

Example: O = [[0, 0], [0, 0]]

7. Upper Triangular Matrix:

A square matrix where all elements below the diagonal are zero.

Example: [[1, 2, 3], [0, 4, 5], [0, 0, 6]]

8. Lower Triangular Matrix:

A square matrix where all elements above the diagonal are zero.

Example: [[1, 0, 0], [2, 3, 0], [4, 5, 6]]

3.3 Matrix Operations

1. Equality of Matrices:

Two matrices A and B are equal if they have the same order and corresponding elements are equal.

A = B if and only if aᵢⱼ = bᵢⱼ for all i, j

2. Addition of Matrices:

If A and B are two matrices of the same order, then their sum A + B is obtained by adding corresponding elements.

(A + B)ᵢⱼ = aᵢⱼ + bᵢⱼ

Example:

A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]]

A + B = [[1+5, 2+6], [3+7, 4+8]] = [[6, 8], [10, 12]]

3. Subtraction of Matrices:

A – B = [[a₁₁-b₁₁, a₁₂-b₁₂], [a₂₁-b₂₁, a₂₂-b₂₂], …]

4. Scalar Multiplication:

If k is a scalar and A is a matrix, then kA is the matrix obtained by multiplying each element of A by k.

kA = [[ka₁₁, ka₁₂], [ka₂₁, ka₂₂], …]

Example: If A = [[1, 2], [3, 4]] and k = 2

2A = [[2, 4], [6, 8]]

5. Matrix Multiplication:

If A is m × n matrix and B is n × p matrix, then their product C = AB is m × p matrix.

cᵢⱼ = Σ(k=1 to n) aᵢₖ × bₖⱼ

Example:

A = [[1, 2], [3, 4]] (2×2) and B = [[5, 6], [7, 8]] (2×2)

AB = [[1×5 + 2×7, 1×6 + 2×8], [3×5 + 4×7, 3×6 + 4×8]]

= [[19, 22], [43, 50]]

3.4 Transpose of a Matrix

The transpose of matrix A, denoted as Aᵀ or A, is obtained by interchanging rows and columns.

Example:

A = [[1, 2, 3], [4, 5, 6]]

Aᵀ = [[1, 4], [2, 5], [3, 6]]

Properties of Transpose:

1. (Aᵀ)ᵀ = A

2. (A + B)ᵀ = Aᵀ + Bᵀ

3. (AB)ᵀ = BᵀAᵀ

4. (kA)ᵀ = k(Aᵀ)

3.5 Symmetric and Skew-Symmetric Matrices

Symmetric Matrix:

A square matrix A is symmetric if A = Aᵀ

This means aᵢⱼ = aⱼᵢ for all i, j

Example: A = [[1, 2, 3], [2, 4, 5], [3, 5, 6]] is symmetric

Skew-Symmetric Matrix:

A square matrix A is skew-symmetric if A = -Aᵀ

This means aᵢⱼ = -aⱼᵢ for all i, j

Note: Diagonal elements must be 0

Example: A = [[0, 2, -3], [-2, 0, 4], [3, -4, 0]] is skew-symmetric

3.6 Solved Examples

Example 1: Find x and y if [[x, 2], [y, 5]] = [[3, 2], [1, 5]]

Solution:

For matrices to be equal, corresponding elements must be equal.

x = 3 and y = 1

Example 2: If A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 2]], find AB and BA

Solution:

AB = [[1×2+2×1, 1×0+2×2], [3×2+4×1, 3×0+4×2]] = [[4, 4], [10, 8]]

BA = [[2×1+0×3, 2×2+0×4], [1×1+2×3, 1×2+2×4]] = [[2, 4], [7, 10]]

Note: AB ≠ BA (matrix multiplication is not commutative)

Example 3: Find Aᵀ if A = [[1, 2, 3], [4, 5, 6]]

Solution:

Aᵀ = [[1, 4], [2, 5], [3, 6]]