Singular and non singular matrix
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1) A singular matrix has a determinant of 0, while a non-singular matrix has a non-zero determinant. 2) A symmetric matrix is equal to its transpose, while a skew-symmetric matrix is equal to the negative of its transpose. 3) The adjoint of a matrix is obtained by swapping diagonal entries and changing the sign of non-diagonal entries. For 3x3 matrices, the adjoint is the transpose of the cofactors.
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A matrix, in general sense, represents a
collection of information stored or arranged
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concept of a matrix refers to a set of numbers,
variables or functions ordered in rows and
columns. Such a set then can be defined as a
distinct entity, the matrix, and it can be
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basic mathematical rules.
A matrix with 9 elements is shown below.
[
[[
[ ]
]]
]
=
==
=
aaa
=
==
=
253
A
131211
aaa
aaa
232221
333231
819
647
Matrix [A] has 3 rows and 3 columns. Each
element of matrix [A] can be referred to by its
row and column number. For example,
=
==
=a
23
6
A computer monitor with 800 horizontal
pixels and 600 vertical pixels can be viewed as
a matrix of 600 rows and 800 columns.
In order to create an image, each pixel is
filled with an appropriate colour.
ORDER OF A MATRIX
The order of a matrix is defined in terms of
its number of rows and columns.
Order of a matrix = No. of rows
×
××
×
No. of
columns
Matrix [A], therefore, is a matrix of order 3
×
××
×
3.
COLUMN MATRIX
A matrix with only one column is called a
column matrix or column vector.
ROW MATRIX
3
6
4
A matrix with only one row is called a row
matrix or row vector.
[
[[
[ ]
]]
]
653
SQUARE MATRIX
A matrix having the same number of rows
and columns is called a square matrix.
742
942
435
RECTANGULAR MATRIX
A matrix having unequal number of rows and
columns is called a rectangular matrix.
1735
13145
8292
REAL MATRIX
A matrix with all real elements is called a real
matrix
PRINCIPAL DIAGONAL and TRACE
OF A MATRIX
In a square matrix, the diagonal containing
the elements a
11
, a
22
, a
33
, a
44
, ……, a
is called
the principal or main diagonal.
The sum of all elements in the principal
diagonal is called the trace of the matrix.
The principal diagonal of the matrix
742
942
435
nn
is indicated by the dashed box. The trace of
the matrix is 2 + 3 + 9 = 14.
UNIT MATRIX
A square matrix in which all elements of the
principal diagonal are equal to 1 while all
other elements are zero is called the unit
matrix.
001
100
010
ZERO or NULL MATRIX
A matrix whose elements are all equal to zero
is called the null or zero matrix.
000
000
000
DIAGONAL MATRIX
If all elements except the elements of the
principal diagonal of a square matrix are
zero, the matrix is called a diagonal matrix.
002
900
030
RANK OF A MATRIX
The maximum number of linearly
independent rows of a matrix [A] is called
the rank of [A] and i.
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6
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ROW MATRIX
3
6
4
A matrix with only one row is called a row
matrix or row vector.
[
[[
[ ]
]]
]
653
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1735
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OF A MATRIX
In a square matrix, the diagonal containing
the elements a
11
, a
22
, a
33
, a
44
, ……, a
is called
the principal or main diagonal.
The sum of all elements in the principal
diagonal is called the trace of the matrix.
The principal diagonal of the matrix
742
942
435
nn
is indicated by the dashed box. The trace of
the matrix is 2 + 3 + 9 = 14.
UNIT MATRIX
A square matrix in which all elements of the
principal diagonal are equal to 1 while all
other elements are zero is called the unit
matrix.
001
100
010
ZERO or NULL MATRIX
A matrix whose elements are all equal to zero
is called the null or zero matrix.
000
000
000
DIAGONAL MATRIX
If all elements except the elements of the
principal diagonal of a square matrix are
zero, the matrix is called a diagonal matrix.
002
900
030
RANK OF A MATRIX
The maximum number of linearly
independent rows of a matrix [A] is called
the rank of [A] and i.
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This document contains 44 true/false statements about determinants and properties of matrices. Some key points:
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The document provides an overview of linear algebra and matrix theory. It discusses the history and development of matrices, defines key matrix concepts like dimensions and operations, and covers foundational topics like matrix addition, multiplication, inverses, and solving systems of linear equations. The document is intended as an introduction to linear algebra and matrices for students.
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Ppt on matrices and Determinants
Matrices can be added, subtracted, and multiplied under certain conditions.
Addition and subtraction require matrices to be the same size.
Matrix multiplication requires the number of columns of the first matrix to equal the number of rows of the second matrix.
Matrices can also be multiplied by scalars.
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The document discusses matrices and their operations. It defines what a matrix is, provides examples of different types of matrices, and covers key matrix operations like addition, subtraction, scalar multiplication, and matrix multiplication. It also defines important matrix concepts such as the transpose of a matrix, inverse of a matrix, and properties related to these operations and concepts.
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Matrices
This document provides an overview of matrices including:
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Determinants
The document discusses determinants and their properties. It defines determinants as representing single numbers obtained by multiplying and adding matrix elements in a special way. It then provides formulas for calculating determinants of matrices of order 1, 2 and 3. It also outlines several properties of determinants, such as how interchanging rows/columns, multiplying rows by constants, and adding rows affects the determinant. Finally, it discusses how determinants are used to determine whether systems of linear equations are consistent or inconsistent.
Determinants
1. Every square matrix can be associated to an expression or a number which is known as its determinant.
Determinants
1. Every square matrix can be associated to an expression or a number which is known as its determinant.
Matrices & Determinants
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Singular and non singular matrix
- 1. Singular and Non Singular Matrix Singular Matrix A Square matrix is Singular if its (mod) || = 0 For example A= 82 41 82 41 = (1)(8) – (2)(4) = 8-8 = 0 Non Singular Matrix A Square matrix is Singular if its (mod) || 0 For Example A= 82 42 = (2)(8)-(2)(4) = 16 – 8 = 8 0 Symmetric
- 2. A Square matrix is symmetric if At = A For Example A= 13 31 At = 13 31 = A Skew Symmetric A Square matrix is symmetric if at = - A For Example A= 04 40 At = 04 40 = - A Harmition Matrix A square matrix with complex entries is called Harmition matrix if |A|t = A
- 3. A= 21 11 i i A = 21 11 i i ( A )t = 21 11 i i = A Skew Harmition Matrix A square matrix with complex entries is called Skew Harmition matrix if |A|t = - A A= 032 320 i i A = 032 320 i i ( A ) t = 032 320 i i = (-1) 032 320 i i = -A Transpose of Matrix Interchanging of rows and columns of any matrix is called transpose (m x n) to (n x m) it is denoted by At
- 4. For Example A = ihg fed cba At = ifc heb gda Determinant For Example A= 82 42 = (2)(8)-(2)(4) = 16 – 8 = Closure Property A + B is a matrix of the same dimensions as A and B. For Example
- 5. A= 52 41 B= 76 34 A + B = 52 41 + 76 34 = 7562 3441 A + B = 128 75 Associative Property A + (B + C) = (A + B) + C A = 76 82 , B = 25 43 , C = 14 21 L. H .S A + (B + C) = 76 82 + ( 25 43 + 14 21 )
- 6. A + (B + C) = 76 82 + ( 1245 2413 ) A + (B + C) = 76 82 + 39 64 A + (B + C) = 3796 6842 A + (B + C) = 1015 146 R.H.S (A + B) + C = ( 76 82 + 25 43 ) + 14 21 (A + B) + C = ( 2756 4832 ) + 14 21 (A + B) + C = 911 125 + 14 21 (A + B) + C = 19411 21215
- 7. (A + B) + C = 1015 146 Which is equal to L.H.S So Associative Proved Commutative A + B = B+A L.H.S A= 52 41 B= 76 34 A + B = 52 41 + 76 34 = 7562 3441 A + B = 128 75 R.H.S
- 8. B + A = 76 34 + 52 41 B + A = 5726 4314 B + A = 128 75 This is equal to L.H.S so commutative property proved Adjoint of 2x2 Matrixes In 2x2 matrix swap the position of diagonal entries and put ( - ) in the front of non diagonal entries It is denoted by Adj( ) A= 52 41 Adj ( A ) = 12 45 Adjoint of 3x3 Matrixes In 3x3 matrix first we find the determinant of matrix and take it transpose
- 9. A = 231 540 213 Adj ( A ) = 333231 232221 1312A11 AAA AAA AA A11 = 23 54 = (4)(2)-(3)(5) = 8 – 15 = -7 A12 = 21 50 = (0)(2)-(1)(5) = 0 – 5 = - 5 A13 = 31 40 = (0)(3)-(1)(4) = 0 – 4 = - 4 A21 = 23 21 = (-1)(2)-(3)(2) = - 2 – 6 = - 8 A22 = 21 23 = (3)(2)-(1)(2) = 6 – 2= 4 A23 = 31 13 = (3)(3)-(-1)(1) = 9 + 1 = 10 A31 = 54 21 = (-1)(5)-(4)(2) = -5 – 8 = - 13 A32 = 50 23 = (3)(5)-(0)(2) = 15 – 0 = 15
- 10. A33 = 40 13 = (3)(4)-(-1)(0) = 12 + 0 = 12 Adj ( A ) = 333231 232221 1312A11 AAA AAA AA = 121513 1048 457 Taking its Transpose Adj (A) = 12154 1545 1387