Mathematics Important topic - Matrix, Study notes of Mathematics

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Matrics

BY-Akansha

Class-XII-D

Roll No.- 03

Content

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Introduction



Elements , Notation Of A

Matrix



Order of a Matrix

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Types Of Matrices

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Operation of Matrices

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Positive Integral Power of

Matrices

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Matrix Polynomial

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Theorem

 Transpose of Matrix

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Symmetric Matrix

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Skew- Symmetric matrix

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Symmetric Matrix and Skew-

Symmetric matrix properties

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Applications of Matrix

Elements , Notation Of A Matrix



Elements

Each of the m.n numbers of m x n matrix is called

elemts or entry of the matric.



Notation

Represented by -[a

ij

] or (a

ij

) or || a

ij

Denoted by single capital letters

Order of a Matrix

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Order

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Matrix with m rows and n columns is said to be order of m x n



A=[a

ij

]

mxn

or A=[a

ij

], i=1,2,3…..m

j= 1,2,3…..n

or A=

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m m ij mn

ij n

ij in

a a a a

a a a a

a a a a

1 2

21 22 2

11 12

...

...

   

Types Of Matrices

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Rectangular Matrix

Matrix is a rectangular matric is m ≠ n

E.g.-

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7 6

7 7

3 7

1 1

Types Of Matrices

Square Matrix

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Matrix is square matrix id m = n

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E.g.-

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The principal or main diagonal of a square matrix is

composed of all elements a

ij

for which i = j

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3 0

1 1

Types Of Matrices

Scalar matrix

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A diagonal matrix whose main diagonal elements are equal

to the same scalar

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A scalar is defined as a single number or constant

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E.g.-

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0 0 1

0 1 0

1 0 0

Types Of Matrices

Unit or Identity matrix – I

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A diagonal matrix with ones on the main diagonal

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E.g.-

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0 1

1 0

Types Of Matrices

Upper triangular matrix

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A square matrix whose elements below the main diagonal

are all zero



E.g.-

Types Of Matrices

Lower triangular matrix

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A square matrix whose elements above the main diagonal

are all zero

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E.g.-

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5 2 3

2 1 0

1 0 0

Types Of Matrices

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Comparable Matrix

 Two matrix A=[a

ij

]

mxn

and B=[b

ij

]

qxr

are comparable only when they are

are of same order i.e only when m= q and n=r.

 E.g.- A= B=



A and B are comparable as they are of same order.

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0 0 3

0 1 8

1 8 7

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6 6 1

9 9 0

1 1 1

Types Of Matrices

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Equality Of Matrices

 Two matrices are said to be equal only when all corresponding elements are equal

 Therefore their size or dimensions are equal as well



E.g.-

A =

B =

A = B

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5 2 3

2 1 0

1 0 0

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5 2 3

2 1 0

1 0 0

Operation of Matrices

Scalar Multiplication of Matrices

Matrices can be multiplied by a scalar (constant or

single element)

Let k be a scalar quantity; then

kA = Ak

Ex. If k=4 and

A

Operation of Matrices

Properties:

k ( A + B ) = k A + k B

(k + g) A = k A + g A

k( AB ) = (k A ) B = A (k) B

k(g A ) = (kg) A

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16 4

8 12

8 4

12 4

4

4 1

2 3

2 1

3 1

4 1

2 3

2 1

3 1

4