PPT: Determinants | Engineering Mathematics - Civil Engineering (CE) PDF Download

 Page 1


DETERMINANTS
• A Determinant of a matrix represents a single 
number. 
• We obtain this value by multiplying and 
adding its elements in a special way. 
Page 2


DETERMINANTS
• A Determinant of a matrix represents a single 
number. 
• We obtain this value by multiplying and 
adding its elements in a special way. 
Determinant of a Matrix of Order One
• Determinant of a matrix of order one 
A=[a
11
]
1x1
is  
?? = a
11
= a
11
Page 3


DETERMINANTS
• A Determinant of a matrix represents a single 
number. 
• We obtain this value by multiplying and 
adding its elements in a special way. 
Determinant of a Matrix of Order One
• Determinant of a matrix of order one 
A=[a
11
]
1x1
is  
?? = a
11
= a
11
Determinant of a Matrix of Order Two
• Determinant of a Matrix A=
?? 11
?? 12
?? 21
?? 22
2x2
is
?? = 
?? 11
?? 12
?? 21
?? 22
= ?? 11
?? 22
-?? 12
?? 21
Page 4


DETERMINANTS
• A Determinant of a matrix represents a single 
number. 
• We obtain this value by multiplying and 
adding its elements in a special way. 
Determinant of a Matrix of Order One
• Determinant of a matrix of order one 
A=[a
11
]
1x1
is  
?? = a
11
= a
11
Determinant of a Matrix of Order Two
• Determinant of a Matrix A=
?? 11
?? 12
?? 21
?? 22
2x2
is
?? = 
?? 11
?? 12
?? 21
?? 22
= ?? 11
?? 22
-?? 12
?? 21
Determinant of a Matrix of Order Three
• Determinant of a Matrix  A = 
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
?? 31
?? 32
?? 33
is  (expanding along R
1
)
• ?? = 
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
?? 31
?? 32
?? 33
= (-1)
1+1
?? 11
?? 22
?? 23
?? 32
?? 33
- (-1)
1+2
?? 12
?? 21
?? 23
?? 31
?? 33
+ (-1)
1+3
?? 13
?? 21
?? 22
?? 31
?? 32
= ?? 11
(?? 22
?? 33
-?? 32
?? 23
) -?? 12
(?? 21
?? 33
-?? 31
?? 23
) +  ?? 13
(?? 21
?? 32
-?? 31
?? 22
) 
= ?? 11
?? 22
?? 33
-?? 11
?? 32
?? 23
-?? 12
?? 21
?? 33
+ ?? 12
?? 31
?? 23
+ ?? 13
?? 21
?? 32
-?? 13
?? 31
?? 22
Page 5


DETERMINANTS
• A Determinant of a matrix represents a single 
number. 
• We obtain this value by multiplying and 
adding its elements in a special way. 
Determinant of a Matrix of Order One
• Determinant of a matrix of order one 
A=[a
11
]
1x1
is  
?? = a
11
= a
11
Determinant of a Matrix of Order Two
• Determinant of a Matrix A=
?? 11
?? 12
?? 21
?? 22
2x2
is
?? = 
?? 11
?? 12
?? 21
?? 22
= ?? 11
?? 22
-?? 12
?? 21
Determinant of a Matrix of Order Three
• Determinant of a Matrix  A = 
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
?? 31
?? 32
?? 33
is  (expanding along R
1
)
• ?? = 
?? 11
?? 12
?? 13
?? 21
?? 22
?? 23
?? 31
?? 32
?? 33
= (-1)
1+1
?? 11
?? 22
?? 23
?? 32
?? 33
- (-1)
1+2
?? 12
?? 21
?? 23
?? 31
?? 33
+ (-1)
1+3
?? 13
?? 21
?? 22
?? 31
?? 32
= ?? 11
(?? 22
?? 33
-?? 32
?? 23
) -?? 12
(?? 21
?? 33
-?? 31
?? 23
) +  ?? 13
(?? 21
?? 32
-?? 31
?? 22
) 
= ?? 11
?? 22
?? 33
-?? 11
?? 32
?? 23
-?? 12
?? 21
?? 33
+ ?? 12
?? 31
?? 23
+ ?? 13
?? 21
?? 32
-?? 13
?? 31
?? 22
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