A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as

1) matrix is enclosed by [ ] or ( ) or | |

2) Compact form the above matrix is represented by

[a

3) Element of a Matrix :The numbers a

4.Order of a Matrix: In above matrix has m rows and n columns, then A is of order m x n

1)

A matrix having only one row and any number of columns is called a row matrix.

2.

A matrix having only one column and any number of rows is called column matrix.

3.

A matrix of order m x n, such that m ≠ n, is called rectangular matrix.

4.

A matrix in which the number of rows is less than the number of columns, is called a horizontal matrix

5.

A matrix in which the number of rows is greater than the number of columns, is called a vertical matrix.

6.

A matrix of any order, having all its elements are zero, is called a null/zero matrix. i.e., aij= 0,∀i,

7)

A matrix of order m x n, such that m = n, is called square matrix.

8.

A square matrix A= [a

9.

A square matrix in which every non-diagonal element is zero and all diagonal elements are equal, is called scalar matrix. i.e., in scalar matrix

a

If

Then

a=1,y=2,z=2

We can only add (or subtract) matrices if they have the same dimensions. That is, the two matrices must have the same number of rows and the same number of columns.

To add matrices, just add corresponding elements:

Commutativity |
A+ B =B+A |

Associativity |
(A+B)+C = A + (B+C) |

Existence of Identity |
A + O = A= O+ AWhere O is the Null matrix |

Existence of Inverse |
A +(-A) =0So -A is the additive inverse of A |

Cancellation |
A+ B= B +CA=C |

Here is the

Here is the

Let

Then Multiplication by scalar would

1) So each element will be multiplied by the scalar

Similary for divison ,every element will be divided by the scalar

Let A and B be m×n matrices, p and q are scalar, thenp(A+B) = pA + qB (p+q)A= pA + qA (pq)A=p(qL) =q(pA) (-p)A= -(pA) = p(-A) |

Two matrics can be multiplied if the number of columns in the first matrix is the same as the number of rows in the second matrix

i.e,

1) 2X3 Matrix can be Multiplied with 3X [1…]

2) 3x2 Matrix can be Multiplied with 2X [1…]

3) 3x2 Matrix can not be Multiplied with [Any number except 2]X [1…]

In general

A =(x,y) And B( y,z) Then Multiplication of A and B is possible, the resultant matric would be C(x,z)

The element of C would be defined as

We can say it this way. We work across the 1st row of the first matrix, multiplying down the 1st column of the second matrix, element by element. We add the resulting products. Our answer goes in position (AB)

We do a similar process for the 1st row of the first matrix and the 2nd column of the second matrix. The result is placed in position (AB)

This is how multiplication works for (2 × 3) and ( 3 ×2) Matrices

AB ≠ BA (AB)C= A(BC) A(B+C)=AB + AC (A+B)C=AC + BC I _{m}A= A= AI_{n} |

Let A =(a

(A

So A

Example

(A ^{T})^{T}=A(A+B) ^{T}=A^{T} + B^{T}(kA) ^{T}=kA^{T}(AB) ^{T}=B^{T}A^{T} |

Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3D model onto a 2 dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.[5] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.

1) You need to remember the rules of addition ,subtraction, multiplication by scalar and Multiplication of Matrix

2) apply the principle and solve the questions

1) Given

Find the negative value of x

My Multiplication of Matrics (1×2) and ( 2×1) ,we get

2x(x) +4(-8) =0

2x

x= 4 or -4

So x =-4

2) Given

Find the Matrix A

Given Matrix equation can be written as

Or

3) Given

Find the value of x and y

**Notes****NCERT Solutions**- NCERT Solution for Matrices Exercise 3.1

Class 12 Maths Class 12 Physics Class 12 Chemistry Class 12 Biology