## Matrices

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, and the general form of a matrix is given by:

In general, we represent the element in row *i* and column *j* as **a _{ij}** .

A matrix having m rows and n columns is called a matrix of **order** m × n or simply m × n matrix (read as an m by n matrix). Further, the number of elements in an m × n matrix will be equal to mn.

## Types of Matrices

#### 1. Row Matrix

A matrix is said to be a **row matrix** if it has only one row. For example:

In general, a row matrix is of the order 1 × n.

#### 2. Column Matrix

A matrix is said to be a **column matrix** if it has only one column. For example:

In general, a column matrix is of the order m × 1.

#### 3. Square Matrix

A matrix in which the number of rows are equal to the number of columns, is said to be a **square matrix**. Thus an m × n matrix is said to be a square matrix if m = n and is known as a square matrix of order ‘n’. For example:

The elements (entries) a_{11}, a_{22}, …, a_{nn} are said to constitute the **diagonal** of a matrix. Here, the diagonal elements of A are: 1, 0, 1.

#### 4. Diagonal Matrix

A square matrix is said to be a **diagonal matrix** if all its non diagonal elements are zero. For example:

#### 5. Scalar Matrix

A diagonal matrix is said to be a **scalar matrix** if its diagonal elements are equal. For example:

#### 6. Identity Matrix

A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an **identity matrix.** We denote the identity matrix of order n by I_{n}. Also, when the order is clear from the context, we simply write it as I. For example:

#### 7. Zero Matrix

A matrix is said to be **zero matrix** or **null matrix** if all its elements are zero. We denote zero matrix by O. For example:

## Equality of Matrices

Two matrices A = [a_{ij}] and B = [b_{ij}] are **equal** if:

1.They are of the same order, and

2.Each element of A is equal to the corresponding element of B, that is a_{ij} = b_{ij} for all *i* and *j*.

## Operations on Matrices

#### Addition of Matrices

In general, if A = [a_{ij}] and B = [b_{ij}] are two matrices of the same order, say m × n. Then, the sum of the two matrices A and B is defined as a matrix C = [c_{ij}]m × n, where c_{ij} = a_{ij} + b_{ij}, for all possible values of i and j. For example:

If A and B are not of the same order, then we cannot define A + B.

#### Properties of Matrix Addition:

1.Commutative Law: A+B=B+A.

2.Associative Law: (A+B)+C=A+(B+C).

3.Additive Identity: A+O=O+A=A.

4.Additive Inverse: A+(-A)=(-A)+A=O.

#### Multiplication of a matrix by a scalar

If A = [a_{ij}] m × n is a matrix and k is a scalar, then kA is another matrix which we obtain by multiplying each element of A by the scalar k. For example:

#### Properties of scalar multiplication of a matrix

1.k(A+B) = kA + kB.

2.(k+p)A = kA+pA.

#### Multiplication of matrices

The product of two matrices A and B exists if the number of columns of A is equal to the number of rows of B.

Let A = [a_{ij}] be an m × n matrix and B = [b_{jk}] be an n × p matrix. Then the product of the matrices A and B is the matrix C of order m × p. To get the (i, k)^{th} element c_{ik} of the matrix C, we take the i^{th} row of A and k^{th} column of B, multiply them elementwise and take the sum of all these products.

If AB is defined, BA need not be defined.

If both A and B are square matrices of the same order, then both AB and BA are defined.

Even if AB and BA are both defined, it is not necessary that AB = BA.

We can also obtain a zero matrix by multiplying two non-zero matrices.

#### Properties of multiplication of matrices

1.Associative Law: (AB)C = A(BC).

2.Distributive Law:

(i) A(B+C) = AB + AC.

(ii) (A+B)C = AC + BC.

3.Multiplicative Identity: IA = AI = A.

## Transpose of a Matrix

If A = [a_{ij}] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A is called the **transpose** of A. We denote transpose of the matrix A by A’. For example:

#### Properties of transpose of the matrices

1.(A’)’ = A.

2.(kA)’ = kA’

3.(A+B)’ = A’ + B’

4.(AB)’ = B’A’

#### Symmetric and Skew Symmetric Matrices

Symmetric Matrix: A is symmetric if A’ = A.

Skew Symmetric Matrix: A is skew symmetric if A’ = -A.

For example:

For any square matrix A with real number entries, A + A’ is a symmetric matrix and A – A’ is a skew symmetric matrix.

We can express any square matrix as the sum of a symmetric and a skew symmetric matrix.

## Elementary Operation (Transformation) of a Matrix

There are six operations (transformations) on a matrix, three of which are due to rows and three due to columns, which are known as elementary operations or transformations.

(i) The interchange of any two rows or two columns. Symbolically the interchange of i^{th} and j^{th} rows is denoted by R_{i}→R_{j} and similarly interchange of i^{th} and j^{th} column is denoted by C_{i}→C_{j}.

(ii) The multiplication of the elements of any row or column by a non zero number. Symbolically, the multiplication of each element of the i^{th} row by k, where k ≠ 0 is denoted by R_{i}→ k R_{i}.

(iii) The addition to the elements of any row or column, the corresponding elements of any other row or column multiplied by any non zero number. Symbolically, the addition to the elements of i^{th} row, the corresponding elements of j^{th} row multiplied by k is denoted by R_{i} →R_{i} + kR_{j}.

#### Invertible Matrices

If A is a square matrix of order m, and if there exists another square matrix B of the same order m, such that AB = BA = I, then we define B as the **inverse** matrix of A and we denote it by A^{– 1}. In that case, we say that A is **invertible.**

The inverse of a square matrix, if it exists, is unique.

If A and B are invertible matrices of the same order, then

(AB)^{– 1} = B^{– 1}A^{– 1}.

#### Inverse of a matrix by elementary operations

Apply row transformation on A=IA to get I=BA, so B is the inverse of A.

Similarly, apply column transformation on A=AI to get I=AB, so B is the inverse of A.

Further, if during these transformations, we obtain all zeros in one or more rows/columns of the matrix on LHS, then A^{– 1} does not exist.

For example:

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