**Matrix:** A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix.

**Order of a Matrix:** If a matrix has m rows and n columns, then its order is written as m × n. If a matrix has order m × n, then it has mn elements.

In general, a_{m×n} matrix has the following rectangular array:

Note: We shall consider only those matrices, whose elements are real numbers or functions taking real values.

**Types of Matrices**

**Column Matrix:** A matrix which has only one column, is called a column matrix.

e.g. \(\left[ \begin{matrix} 1 \\ 0 \\ -5 \end{matrix} \right]\)

In general, A = [a_{ij}]_{m×1} is a column matrix of order m × 1.

**Row Matrix:** A matrix which has only one row, is called a row matrix,

e.g. \(\left[ \begin{matrix} 1 & 5 & 9 \end{matrix} \right]\)

In general, A = [a_{ij}]_{1×n} is a row matrix of order 1 x n

**Square Matrix:** A matrix which has equal number of rows and columns, is called a square matrix

e.g. \(\begin{bmatrix} 3 & -1 \\ 5 & 2 \end{bmatrix}\)

In general, A = [a_{ij}]m x m is a square matrix of order m.

Note: If A = [a_{ij}] is a square matrix of order n, then elements a_{11}, a_{22}, a_{33},…, a_{nn }is said to constitute the diagonal of the matrix A.

**Diagonal Matrix:** A square matrix whose all the elements except the diagonal elements are zeroes, is called a diagonal matrix,

e.g. \(\left[ \begin{matrix} 3 & 0 & 0 \\ 0 & -3 & 0 \\ 0 & 0 & -8 \end{matrix} \right]\)

In general, A = [a_{ij}]_{m×m} is a diagonal matrix, if a_{ij} = 0, when i ≠ j.

**Scalar Matrix:** A diagonal matrix whose all diagonal elements are same (non-zero), is called a scalar matrix,

e.g. \(\left[ \begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{matrix} \right]\)

In general, A = [a_{ij}]_{n×n} is a scalar matrix, if a_{ij} = 0, when i ≠ j, a_{ij} = k (constant), when i = j.

Note: A scalar matrix is a diagonal matrix but a diagonal matrix may or may not be a scalar matrix.

**Unit or Identity Matrix:** A diagonal matrix in which all diagonal elements are ‘1’ and all non-diagonal elements are zero, is called an identity matrix. It is denoted by I.

e.g. \(\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right]\)

In general, A = [a_{ij}]_{n×n} is an identity matrix, if a_{ij} = 1, when i = j and a_{ij} = 0, when i ≠ j.

**Zero or Null Matrix:** A matrix is said to be a zero or null matrix, if its all elements are zer0

e.g. \(\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\)

**Equality of Matrices:** Two matrices A and B are said to be equal, if

(i) order of A and B are same.

(ii) corresponding elements of A and B are same i.e. a_{ij} = b_{ij}, ∀ i and j.

e.g. \(\begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}\) and \(\begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}\) are equal matrices, but \(\begin{bmatrix} 3 & 2 \\ 0 & 1 \end{bmatrix}\) and \(\begin{bmatrix} 2 & 3 \\ 0 & 1 \end{bmatrix}\) are not equal matrices.

**Operations on Matrices**

Between two or more than two matrices, the following operations are defined below:

**Addition and Subtraction of Matrices:** Addition and subtraction of two matrices are defined in an order of both the matrices are same.

Addition of Matrix

If A = [a_{ij}]_{m×n} and B = [y_{ij}]_{m×n}, then A + B = [a_{ij} +b_{ij}]_{m×n}, 1 ≤ i ≤ m, 1 ≤ j ≤ n

Subtraction of Matrix

If A = [a_{ij}]_{m×n} and B = [b_{ij}]_{m×n}, then A – B = [a_{ij} – b_{ij}]_{m×n}, 1 ≤ i ≤ m, 1 ≤ j ≤ n

**Properties of Addition of Matrices**

(a) Commutative If A = [a_{ij}] and B = [b_{ij}] are matrices of the same order say m x n then A + B = B + A,

(b) Associative for any three matrices A = [a_{ij}], B = [b_{ij}], C = [c_{ij}] of the same order say m x n, A + (B + C) = (A + B) + C.

(c) Existence of additive identity Let A = [aij] be amxn matrix and O be amxn zero matrix, then A + O = O + A = A. In other words, O is the additive identity for matrix addition.

(d) Existence of additive inverse Let A = [a_{ij}]_{m×n} be any matrix, then we have another matrix as -A = [-a_{ij}]_{m×n} such that A + (-A) = (-A + A) = O. So, matrix (-A) is called additive inverse of A or negative of A.

Note

(i) If A and B are not of the same order, then A + B is not defined.

(ii) Addition of matrices is an example of a binary operation on the set of matrices of the same order.

**Multiplication of a matrix by scalar number:** Let A = [a_{ij}]_{m×n} be a matrix and k is scalar, then kA is another matrix obtained by multiplying each element of A by the scalar k, i.e. if A = [a_{ij}]_{m×n}, then kA = [ka_{ij}]_{m×n}.

Properties of Scalar Multiplication of a Matrix

Let A = [a_{ij}] and B = [b_{ij}]be two matrices of the same order say m × n, then

(a) k(A + B) = kA + kB, where k is a scalar.

(b) (k + l)A = kA + lA, where k and l are scalars.

**Multiplication of Matrices:** Let A and B be two matrices. Then, their product AB is defined, if the number of columns in matrix A is equal to the number of rows in matrix B.

Properties of Multiplication of Matrices

(a) Non-commutativity Matrix multiplication is not commutative i.e. if AB and BA are both defined, then it is not necessary that AB ≠ BA.

(b) Associative law For three matrices A, B, and C, if multiplication is defined, then A (BC) = (AB) C.

(c) Multiplicative identity For every square matrix A, there exists an identity matrix of the same order such that IA = AI = A.

Note: For Amxm, there is only one multiplicative identity I_{m}.

(d) Distributive law For three matrices A, B, and C,

A(B + C) = AB + AC

(A + B)C = AC + BC

whenever both sides of the equality are defined.

Note: If A and B are two non-zero matrices, then their product may be a zero matrix.

e.g. Suppose A = \(\begin{bmatrix} 0 & -1 \\ 0 & 2 \end{bmatrix}\) and B = \(\begin{bmatrix} 3 & 5 \\ 0 & 0 \end{bmatrix}\), then AB = \(\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\).