# Matrices Class 12 Notes Maths Chapter 3 - CBSE

## What are 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.

• The horizontal lines of elements are said to be, rows of the matrix.
• The vertical lines of elements are said to be, columns of the matrix.

## Order Of Matrix

A matrix having m rows and n columns is called a matrix of order m × n.

• Namely A = [aij]m×n to indicate that A is a matrix of order m × n.

## Equality Of Matrix

Two matrices A = [aij] and B = [bij] are said to be equal if :

• They are of the same order.
• Each element of A is equal to the corresponding element of B , that is aij = bij for all i and j .

## Types Of Matrices

Row Matrix

A matrix is said to be a row matrix if it has only one row.

Square Matrix

A matrix in which the number of rows are equal to number of columns.

Diagonal Matrix

A square matrix is said to be a diagonal matrix if all its non-diagonal elements are zero.

Scalar Matrix

A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal.

Zero Matrix

A matrix is said to be zero matrix or null matrix if all its elements are zero.

Identity Matrix

A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix.

Column Matrix

A matrix is said to be column matrix if it has only one column.

## Transpose Of A Matrix

The transpose of a matrix is found by interchanging its row into columns or columns into rows. The transpose of matrix is represented by A' or AT.

## Symmetric And Skew Symmetric Matrices

A symmetric and skew symmetric matrix both are square matrices. The symmetric matrix is equal to its transpose. The skew symmetric matrix is a matrix whose transpose is equal to its negative.

## Operations On Matrices

There are certain operations on matrices, addition of matrices, multiplication of matrix by scalar and multiplication of matrices.

If A = [aij] and B = [bij] 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 = [cij]m × n , where cij = aij + bij for all possible values of i and j .

Multiplication of matrices

The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B. Let A = [aij] be an m × n matrix and B = [bjk] be an n × p matrix. Then the product of the matrices A and B is the matrix C of order m × p.

Multiplication of a matrix by scalar

If A = [aij]m×n is a matrix and k is a scalar, then kA is another matrix which is obtained by multiplying each element of A by the scalar k.

## Properties

• Commutative Law: If A = [aij] and B = [bij] are two matrices of the same order, say m × n, then A + B = B + A.
• Associative Law: If A=[aij], B=[bij] and C=[cij] are three matrices of the same order, say m × n, then (A + B) + C = A + (B + C).
• Additive Identity: Let A = [aij] be an m × n matrix and O be an m × n zero matrix, then A + O = O + A = A.
• Additive Inverse: Let A = [aij]m×n be any matrix, then we have another matrix as -A = [-aij]m×n such that A + (-A) = (-A) + A = O.

Properties of multiplication of matrices

• Associative Law: For any three matrices A,B and C. We have (AB)C = A(BC).
• The Distribution Law: For any three matrices A,B and C. A(B+C) = AB + AC and (A+B)C = AC + BC
• Existence of Multiplicative Identity: For every square matrix A, there exist an identity matrix of same order such that IA = AI = A.
• Matrix multiplication is not commutative AB BA

Properties of scalar multiplication of a matrix

• k(A+B) = kA + kB, where k is scalar.
• (k+l)A = kA + lA, where k and l are scalars.

## 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 B is called the inverse matrix of A and denoted by A-1
• Inverse of a square matrix, if it exists is unique.