## What is matrices ?

**Defnition: **A set of **mn** numbers (**real or imaginary**) arranged in the form of **a rectangular array of m rows and n columns is called an** **m × n **(to be read as ‘m by n’ matrices).

An m×n matrix is usually written as

A compact form the above matrix is represented by or .

### Some points related to Matrices

**(1)** The numbers etc known as the elements of the matrices A.

**(2)** The element belongs to ith row and jth column and is called the (i,j)th element of the matrix .

**(3)** In the element the first subscript **i,** always denotes the number of rows and the second subscript** j**, denotes the number of columns.

**Following are some examples of matrices:**

**(i)** is a matrix having 3 rows and 2 columns and so it is a matrix of order 3 × 2 such that .

**(ii)** is a matrix having 2 rows and 2 columns and so it is a matrix of order 2 × 2 such that .

## Types of matrices

### Row Matrix:(What is Row Matrix)

A matrix having only one row is called a row-matrix or a row vector .

**For example**, is a row matrix of order 1 × 4.

### Column Matrix:(What is Column Matrix)

A matrix having only one column is called a column-matrix or a column-vector.

**For example**, is a column matrix of order 4 × 1.

### Square matrix:(What is Scalar Matrix)

A matrix in which the number of rows is equal to the number of columns,say n, is called a square matrix of order n.

**For example**, The matrix is a square matrix of order 3

### Diagonal Matrix:(What is Diagonal matrix ?)

A square matrix is called a diagonal matrix if all the elements, except those in the leading diagonal, are zero i.e.

If a diagonal matrix of order 3 × 3 having a,b,c as diagonal elements is denoted by **diag[a, b, c].**

**Example:** The matrix is a diagonal matrix, to be denoted by **A = diag[1, 2, 3]**

### Scalar Matrix:( What is scalar matrix ?)

A squarte matrix is called a scalar matrix if

**(i)** , and

**(ii)** where C ≠ 0

In other words, a diagonal matrix in which all the diagonal elements are equal is called the scalar matrix.

**Example: ** The matrices and are scalar matrices of order 3 and 2 respectively.

### Identity or Unit matrix:( What is Identity or Unit Matrix)

A squarte matrix is called an identity or unit matrix if

**(i)** , and

**(ii)** where C ≠ 0

In other words, a square matrix each of whose diagonal elements is unity and each of whose non-diagonal elements is equal zero is called an identity or Unit matrix.

Identity matrix of order n is denoted by .

**Example: ** The matrices and are Identity matrices of order 3 and 2 respectively.

### NULL MATRIX: (What is Null Matrix)

A Matrix whose all elements are zero is called a Null Matrix or Zero Matrix.

Example: are null matrices of order 2×2 and 2×3 respectively.

### UPPER TRIANGULAR MATRIX:

A square matrix is called an Upper Triangular Matrix if For all i>j.

Thus, in an Upper Triangular Matrix, all elements below the main diagonal are zero.

**Example:** is an Upper Triangular Matrix.

### LOWER TRIANGULAR MATRIX:

A square matrix is called an Upper Triangular Matrix if For all i<j.

Thus, in an Lower Triangular Matrix, all elements above the main diagonal are zero.

**Example:** is an Lower Triangular Matrix.

### Strictly Triangular Matrix:

A Triangular Matrix is called a strictly triangular matrix for all i = 1, 2, . . . ,n.

## Equality Of Matrices

Two matrices are equal if

**(i)** m = r, i.e., the number of rows in A equal the number of rows in B.

**(ii)** n = s, i.e., The number of column in A equals the number of column in B.

**(iii)** for i = 1, 2, . . ., m and j = 1, 2, . . ., n

If two matrices A and B are equal, we write A = B, otherwise we write A ≠ B.

**Example 1:-** If find x, y, z, w.

**Solution:** Since the corresponding elements of two equal matrices are equal, therefore

⇒ x – y = -1, 2x – y = 0, 2x + z = 5, 3z + w = 13.

Solving the equation and we get

x = 1, y = 2, z = 3, anw w = 4.

**Example 2:-** Matrices and are not equal, because their order are not same.

## Algebra Of Matrices:

Let A, B be two matrices, each of order **m×n. **Then their sum A + B is a matrix of order **m×n **and is obtained by adding the corresponding elements of A and B.

Thus, if and are two matrices of the same order, their sum A + B is defined to be the matrix of order **m×n **such that

for i = 1, 2, . . , m and j = 1, 2, . . ., n

**NOTE:-** The sum of two matrices is defined only when they are of the same order.

**Example:-** If , then

### Properties Of Matrix Addition

**(i) Matrix addition is commutative**

A + B = B + A

**(ii) Matrix addition is associative**

A + (B + C) = (A + B) + C

**(iii) Existence of Identity**

A + O = A = O + A

The Null matrix is the identity matrix for matrix addition and it is denoted by **O**

**(iv) Existence of Inverse**

A + (-A) = O = (-A) + A

Inverse of matrix **A** is **-A**

**(v) Cancellation laws hold good in case of addition of matrices**

A + B = A + C ⇒ B = C

and B + A = C + A ⇒ B = C

## Multiplication of a matrix by a scalar(Scalar multiplication)

Let be an m×n matrix and **k** be any number called a scalar. Then the matrix obtained by multiplication every element of **A** by **k** is called the scalar multiple of **A** by **k** and is denoted by **kA**. Thus

**Example:-** If

### Properties of scalar multiplication

If are two matrices and k, l are scalars, then

**(i)** k(A + B) = kA + kB

**(ii)** (k + l)A = kA + lA

**(iii)** (kl)A = k(lA) = l(kA)

**(iv)**(-k)A = -(kA) = k(-A)

**(v)** 1A = A

**(vi)** (-1)A = -A.

## Subtraction Of Matrices(Definition)

For two matrices A and B of the same order, we define A – B = A + (-B).

**Example:- **If , then

## Multiplication Of Matrices

Two matrices A and B are conformable for the product AB if **the number of column in A(Post multiplier) is same as the number of rows in B(Pre multiplier)**. Thus, if are two matrices of order m×n and n×p respectively, then **their product AB is of order m×p **

Let matrix of order 1×3 and is a matrix of order 3×1

**The column of matrix A is 3 and the row of matrix B is also 3, then the AB is defined**

Now,

= ap + bq + cr

**NOTE:- If A and B are two matrices such that AB exists, then BA may or may not exist.**

### Properties Of Matrix Multiplication

**(i) Matrix multiplication is not commutative in genral.**

AB ≠ BA

**(ii) Matrix multiplication is associative i.e.**

(AB)C = A(BC), Whenever both sides are defined

**(iii) Matrix multiplication is distributive over matrix addition**

i.e. **(a)** A(B + C) = AB + AC

**(b)** (A + B)C = AC + AC

Whenever both sides of equality are defined.

**(iv) If A is an m×n matrix, then**

**(v) The product of two matrices can be the null matrix while neither of them is the null matrix.**

**Example:-** Let

Then, while neither A nor B is the **null matrix**.

**(vi) If A is m×n matrix and o is a null matrix, then**

**(a)**

**(b)**

**(vii) In case of matrix multiplication if AB = O, then it does not necessarily imply that BA = O.**

**Example:- **Let . Then AB = O. But

≠ O

Thus, AB = O while BA ≠ O.

### Positve Integral Power of a Square Matrix

Let A be a square matrix. Then we define

(i) A¹ = A

(ii)

(iii) for all m, n ∈ N

## Matrix Polynomial

Let be a polynomial and let A be a square matrix of order n. Then

is called a polynomial matrix.

## Question related to matrix

**Question 1:** Matrix Theory introduced by ………..

**Answer:** Matrix theory introduced by **Cayley-Hamilton .**

**Question 2:** Can a diagonal matrix is both an upper triangular and lower triangular matrix ?

**Answer:** **Yes**, **a diagonal matrix is both an upper triangular and lower triangular matrix **

**Question 4:** If S is a scalar matrix and A is a square matrix of the same order n, then AS = SA is true or false .

**Answer :** **AS = SA is true for the statement.**

**Question 5:** If A and B are two matrices such that AB and A + B are both defined, then A, B are square matrices of different order .

**Answer:** **No, A and B matrices should be of same order**

### Some Other Solution

Class 12 ncert solution math exercise 3.1 matrix

Class 12 ncert solution math exercise 3.2 matrix

class 12 maths exercise 3.3 solution

class 12 maths chapter 3 Miscellaneous solution