MATRIX

Definition

A matrix is a rectangular arrangement of numbers, symbols, or expressions in rows and columns. The numbers or symbols in a matrix are called its elements or entries. If a matrix has m rows and n columns, it is called an m × n (read as “m by n”) matrix.

Types of Matrices

• Row Matrix: A matrix with only one row.
• Column Matrix: A matrix with only one column.
• Square Matrix: Number of rows = number of columns (m = n).
• Diagonal Matrix: A square matrix with all non-diagonal elements zero.
• Scalar Matrix: A diagonal matrix with all diagonal elements equal.
• Identity Matrix: A diagonal matrix with all diagonal elements equal to 1.
• Zero Matrix: All elements are zero.

Example

Let
A = (\begin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix})
This is a 2 × 2 matrix.

Matrix Notation

A matrix is usually denoted by a capital letter (A, B, C, etc.). The element in the i-th row and j-th column is denoted by (a_{ij}).

Basic Operations

1. Addition and Subtraction

If A and B are matrices of the same order, then
A + B = [a_{ij} + b_{ij}]
A – B = [a_{ij} – b_{ij}]

2. Scalar Multiplication

If k is a scalar and A = [a_{ij}], then
kA = [k × a_{ij}]

3. Matrix Multiplication

If A is an m × n matrix and B is an n × p matrix, then their product AB is an m × p matrix.
(AB){ij} = Σ (a{ik} × b_{kj}) for k = 1 to n

4. Transpose

The transpose of matrix A, denoted by A^T, is obtained by interchanging rows and columns.
If A = [a_{ij}], then A^T = [a_{ji}]

5. Determinant (for 2 × 2 matrix)

If A = (\begin{bmatrix} a & b \ c & d \end{bmatrix}),
then |A| = ad – bc

6. Inverse (for 2 × 2 matrix)

If A = (\begin{bmatrix} a & b \ c & d \end{bmatrix}),
and |A| ≠ 0, then
A^{-1} = (1/|A|) × (\begin{bmatrix} d & -b \ -c & a \end{bmatrix})

Example Calculations

Let
A = (\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}),
B = (\begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix})

• Addition:
  A + B = (\begin{bmatrix} 1+5 & 2+6 \ 3+7 & 4+8 \end{bmatrix}) = (\begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix})
• Scalar Multiplication:
  2A = (\begin{bmatrix} 2×1 & 2×2 \ 2×3 & 2×4 \end{bmatrix}) = (\begin{bmatrix} 2 & 4 \ 6 & 8 \end{bmatrix})
• Matrix Multiplication:
  AB = (\begin{bmatrix} (1×5+2×7) & (1×6+2×8) \ (3×5+4×7) & (3×6+4×8) \end{bmatrix})
  = (\begin{bmatrix} 19 & 22 \ 43 & 50 \end{bmatrix})
• Transpose:
  A^T = (\begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix})
• Determinant:
  |A| = (1×4) – (2×3) = 4 – 6 = -2
• Inverse:
  A^{-1} = (1/(-2)) × (\begin{bmatrix} 4 & -2 \ -3 & 1 \end{bmatrix})
  = (\begin{bmatrix} -2 & 1 \ 1.5 & -0.5 \end{bmatrix})

Summary Table

Operation	Formula/Result
Addition	A + B = [a_{ij} + b_{ij}]
Scalar Multiplication	kA = [k × a_{ij}]
Matrix Multiplication	(AB){ij} = Σ (a{ik} × b_{kj})
Transpose	A^T = [a_{ji}]
Determinant (2×2)	ad – bc

About The Author