PPT: Determinants

# 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
• http://www.authorstream.com/Presentation/j
oshsmith1110-162211-determinants-math-
ppt-shella-paglinawan-education-powerpoint/
```

## Engineering Mathematics

65 videos|121 docs|94 tests

## FAQs on PPT: Determinants - Engineering Mathematics - Civil Engineering (CE)

 1. What are determinants in mathematics?
Ans. Determinants are mathematical tools used to analyze square matrices. They provide information about the matrix's properties, such as its invertibility and solutions to systems of linear equations.
 2. How can determinants be calculated?
Ans. Determinants can be calculated using various methods, such as expansion by minors, cofactor expansion, or using properties of determinants like row operations or column operations. The specific method used depends on the size and properties of the matrix.
 3. What is the significance of determinants in linear algebra?
Ans. Determinants play a crucial role in linear algebra as they provide information about the linear independence of vectors, the existence and uniqueness of solutions to systems of linear equations, and the invertibility of matrices.
 4. Can determinants be negative?
Ans. Yes, determinants can be negative. The sign of a determinant depends on the arrangement of its elements and can be positive, negative, or zero. The negative sign indicates the matrix's orientation changes during certain transformations.
 5. How are determinants used in solving systems of linear equations?
Ans. Determinants are used to determine whether a system of linear equations has a unique solution, no solution, or infinitely many solutions. By evaluating the determinant of the coefficient matrix, one can determine the system's solvability and find the solutions if they exist.

## Engineering Mathematics

65 videos|121 docs|94 tests

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