Definition:

A matrix is a two-dimensional array of numbers or expressions arranged in rows and columns. An m × n matrix A has m rows and n columns and is written as:

$$ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} $$

where the element \( a_{ij} \) is located in the \( i^{th} \) row and the \( j^{th} \) column.

Notations:
\( \mathbb{M}_{m \times n} (\mathbb{R}) \) – the set of all \( m \times n \) matrices with real numbers as its elements.

Special Matrices

  1. Square Matrix:

If the number of rows and the number of columns of a matrix are equal, then it is called a square matrix.

  1. Zero Matrix:

A zero matrix or null matrix is a matrix with all elements equal to zero.

  1. Diagonal Matrix:

A square matrix is said to be diagonal if each one of the non-diagonal entries is zero.

  1. Scalar Matrix:

A diagonal matrix in which all the diagonal entries are equal is said to be a scalar matrix.

  1. Column Vector or Column Matrix:

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

  1. Row Vector or Row Matrix:

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

  1. Upper Triangular Matrix:

A square matrix is said to be an upper triangular matrix if all its entries below the main diagonal are zero.

  1. Strictly Upper Triangular Matrix:

If the entries on the main diagonal of an upper triangular matrix are all zero, the matrix is said to be strictly upper triangular.

  1. Lower Triangular Matrix:

A square matrix is said to be a lower triangular matrix if all its entries above the main diagonal are zero.

  1. Strictly Lower Triangular Matrix:

If the entries on the main diagonal of a lower triangular matrix are all zero, the matrix is said to be strictly lower triangular.

  1. Identity Matrix:

A square matrix in which all the main diagonal elements are 1’s and all the remaining elements are 0’s is called an identity matrix.

Equality of Matrices

Two matrices are equal if they have the same size and the same corresponding entries.

Operations on Matrices

  • Addition of Matrices: The addition of A and B is denoted by \( A + B \), and is defined as \( A + B = [a_{ij} + b_{ij}]_{m \times n} \).
  • Subtraction of Matrices: The subtraction of A and B is denoted by \( A - B \), and is defined as \( A - B = [a_{ij} - b_{ij}]_{m \times n} \).
  • Scalar Multiplication: The scalar multiplication of A with a real number c is denoted by \( cA \) and is defined as \( cA = [ca_{ij}]_{m \times n} \).

Theorem 1.1

Let \( \mathbb{M}_{m \times n} (\mathbb{R}) \) be the collection of all \( m \times n \) matrices over the field of real numbers. If A, B, C \( \in \mathbb{M}_{m \times n} (\mathbb{R}) \) and \( \alpha, \beta \in \mathbb{R} \), then:

  1. \( A + B = B + A \)
  2. \( A + (B + C) = (A + B) + C \)
  3. \( A + 0 = A \)
  4. \( A + (-A) = 0 \)
  5. \( (\alpha + \beta)A = \alpha A + \beta A \)
  6. \( \alpha (A + B) = \alpha A + \alpha B \)
  7. \( (\alpha \beta) A = \alpha (\beta A) \)
  8. \( 1A = A \)

These properties show that the set \( \mathbb{M}_{m \times n} (\mathbb{R}) \) of all \( m \times n \) matrices is a vector space over the field of real numbers.

References

  • Lay, D.C., Lay, S.R., & McDonald, J.J. (2016). Linear Algebra and its Applications (5th ed.). Pearson.