12.1 Definition
Every square matrix can be associated to an expression or a number which is known as determinant.
If the matrix has only one element a11 then a11 is the determinant.
If the matrix is of order 2 that 2 by 2 matrix
|A| =
|a11 a12|
|a21 a22| =
a11*a22 – a12*a21
Determinant of a square matrix of order 3
Determinant of a square matrix of order 3 is the sum of the product of elements a1j in the first row with (-1) 1+j times the determinant of a 2×2 sub-matrix obtained by leaving the first row and column passing through the element.
(i) Only square matrices have determinants.
(ii) The determinant of a square matrix of order three can be expanded along any row or column.
Determinant of a square matrix of order 4 or more
(iii) Determinant of a square matrix of order 4 or more can be determined following the procedure of finding the determinant of a square matrix of order 3. But in this case, especially in the case of 4×4 matrix, when we omit the rows and columns containing the elements of a row, we get 3×3 sub-matrices and we have to find determinants for them.
12.2 Singular matrix
A square matrix is a singular matrix if its determinant is zero.
Otherwise it is a non-singular matrix.
12.3 Minors and cofactors
Minor: For a square matrix [aij] or order n, the minor Mij of aij, in A is the determinant of the square sub-matrix of order (n-1), obtained by leaving (or striking off) ith row and jth column of A.
Cofactor: Cofactor of an element aij in a square matrix [aij] is termed Cij.
Cij = (-1) i+j Mij
Mij is the minor of element aij in a square matrix [aij].
Minors and cofactors are defined for elements of a square matrix only. They are not defined for determinants.
12.4 Properties of determinants
1. For a square matrix, the sum of the product of elements of any row (or column) with their cofactors is always equal to determinants of the matrix.
2. For a square matrix, the sum of the product of elements of any row (or column) with the cofactors of corresponding elements of some other row (or column) is zero.
3. The value of a determinant remains unchanged if its rows and columns are interchanged.
4. If any two rows (or columns) of a determinant are interchanged, then the value of the determinant changes by minus sign only.
5. If any two rows or columns of a determinant are identical then its value is zero.
6. If each element of a row (or a column) of a determinant is multiplied by a constant k, then the value of the new determinant is ‘k’ times the value of the original determinant.
7. If each element of a row (or a column) of a determinant is expressed as a sum of two or more terms, then the determinant can be expressed as the sum of two or more determinants of the same order.
8. If each element of a row (or a column) of a determinant is multiplied by the same constant and then added to the corresponding elements of some other row (column) then the value of the determinant remains same.
9. If each element of a row (or column) in a determinant is zero, then its value is zero.
10. If the matrix is a diagonal square matrix then its determinant is the product of all the diagonal elements.
11. If A and B are square matrices of the same order, then
|AB| = |A| |B|
12. If a matrix is a triangular matrix of order n, then its determinant is the product of all the diagonal elements.
12.5 Evaluation of Determinants
To evaluate determinants or large matrices, we use the properties of determinants given in section 12.4 above, to create many zeroes in the elements of a row or column and then expand the determinant using elements and cofactors of that row or column.
12.6 Evaluation of Determinants by using Factor Theorem
If f(x) is a polynomial and f(α) = 0 the, (x- α) is a factor of f(x).
If a determinant is a polynomial in x, then (x- α) is factor of the determinant if its value is zero when we put x = α.
Using this rule we can find determinant as a product of its factors.
12.7 Product of Determinants
A definition of product of determinants is similar to the rule of multiplication of matrices.
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Updated 17 Jan 2016, 7 June 2008
Showing posts with label Determinants. Show all posts
Showing posts with label Determinants. Show all posts
Sunday, January 17, 2016
Tuesday, December 9, 2008
Determinants - Definition
Every square matrix can be associated to an expression or a number which is known as determinant.
If the matrix has only one element a11 then a11 is the determinant.
If the matrix is of order 2 that 2 by 2 matrix
|A| =
|a11 a12|
|a21 a22| =
a11*a22 – a12*a21
Determinant of a square matrix of order 3
Determinant of a square matrix of order 3 is the sum of the product of elements a1j in the first row with (-1) 1+j times the determinant of a 2×2 sub-matrix obtained by leaving the first row and column passing through the element.
(i) Only square matrices have determinants.
(ii) The determinant of a square matrix of order three can be expanded along any row or column.
Determinant of a square matrix of order 4 or more
(iii) Determinant of a square matrix of order 4 or more can be determined following the procedure of finding the determinant of a square matrix of order 3. But in this case, especially in the case of 4×4 matrix, when we omit the rows and columns containing the elements of a row, we get 3×3 sub-matrices and we have to find determinants for them.
If the matrix has only one element a11 then a11 is the determinant.
If the matrix is of order 2 that 2 by 2 matrix
|A| =
|a11 a12|
|a21 a22| =
a11*a22 – a12*a21
Determinant of a square matrix of order 3
Determinant of a square matrix of order 3 is the sum of the product of elements a1j in the first row with (-1) 1+j times the determinant of a 2×2 sub-matrix obtained by leaving the first row and column passing through the element.
(i) Only square matrices have determinants.
(ii) The determinant of a square matrix of order three can be expanded along any row or column.
Determinant of a square matrix of order 4 or more
(iii) Determinant of a square matrix of order 4 or more can be determined following the procedure of finding the determinant of a square matrix of order 3. But in this case, especially in the case of 4×4 matrix, when we omit the rows and columns containing the elements of a row, we get 3×3 sub-matrices and we have to find determinants for them.
Singular matrix
A square matrix is a singular matrix if its determinant is zero.
Otherwise it is a non-singular matrix.
Otherwise it is a non-singular matrix.
Minors and cofactors
Minor: For a square matrix [aij] or order n, the minor Mij of aij, in A is the determinant of the square sub-matrix of order (n-1), obtained by leaving (or striking off) ith row and jth column of A.
Cofactor: Cofactor of an element aij in a square matrix [aij] is termed Cij.
Cij = (-1)i+jM)ij
Mij is the minor of element aij in a square matrix [aij].
Minors and cofactors are defined for elements of a square matrix only. They are not defined for determinants.
Cofactor: Cofactor of an element aij in a square matrix [aij] is termed Cij.
Cij = (-1)i+jM)ij
Mij is the minor of element aij in a square matrix [aij].
Minors and cofactors are defined for elements of a square matrix only. They are not defined for determinants.
Properties of determinants
1. For a square matrix, the sum of the product of elements of any row (or column) with their cofactors is always equal to determinants of the matrix.
2. For a square matrix, the sum of the product of elements of any row (or column) with the cofactors of corresponding elements of some other row (or column) is zero.
3. The value of a determinant remains unchanged if its rows and columns are interchanged.
4. If any two rows (or columns) of a determinant are interchanged, then the value of the determinant changes by minus sign only.
5. If any two rows or columns of a determinant are identical then its value is zero.
6. If each element of a row (or a column) of a determinant is multiplied by a constant k, then the value of the new determinant is ‘k’ times the value of the original determinant.
7. If each element of a row (or a column) of a determinant is expressed as a sum of two or more terms, then the determinant can be expressed as the sum of two or more determinants of the same order.
8. If each element of a row (or a column) of a determinant is multiplied by the same constant and then added to the corresponding elements of some other row (column) then the value of the determinant remains same.
9. If each element of a row (or column) in a determinant is zero, then its value is zero.
10. If the matrix is a diagonal square matrix then its determinant is the product of all the diagonal elements.
11. If A and B are square matrices of the same order, then
|AB| = |A| |B|
12. If a matrix is a triangular matrix of order n, then its determinant is the product of all the diagonal elements.
2. For a square matrix, the sum of the product of elements of any row (or column) with the cofactors of corresponding elements of some other row (or column) is zero.
3. The value of a determinant remains unchanged if its rows and columns are interchanged.
4. If any two rows (or columns) of a determinant are interchanged, then the value of the determinant changes by minus sign only.
5. If any two rows or columns of a determinant are identical then its value is zero.
6. If each element of a row (or a column) of a determinant is multiplied by a constant k, then the value of the new determinant is ‘k’ times the value of the original determinant.
7. If each element of a row (or a column) of a determinant is expressed as a sum of two or more terms, then the determinant can be expressed as the sum of two or more determinants of the same order.
8. If each element of a row (or a column) of a determinant is multiplied by the same constant and then added to the corresponding elements of some other row (column) then the value of the determinant remains same.
9. If each element of a row (or column) in a determinant is zero, then its value is zero.
10. If the matrix is a diagonal square matrix then its determinant is the product of all the diagonal elements.
11. If A and B are square matrices of the same order, then
|AB| = |A| |B|
12. If a matrix is a triangular matrix of order n, then its determinant is the product of all the diagonal elements.
Evaluation of Determinants
To evaluate determinants or large matrices, we use the properties of determinants, to create many zeroes in the elements of a row or column and then expand the determinant using elements and cofactors of that row or column.
Evaluation of Determinants by using Factor Theorem
If f(x) is a polynomial and f(α) = 0 the, (x- α) is a factor of f(x).
If a determinant is a polynomial in x, then (x- α) is factor of the determinant if its value is zero when we put x = α.
Using this rule we can find determinant as a product of its factors.
If a determinant is a polynomial in x, then (x- α) is factor of the determinant if its value is zero when we put x = α.
Using this rule we can find determinant as a product of its factors.
Product of Determinants
A definition of product of determinants is similar to the rule of multiplication of matrices.
Differentiation of Determinants
If Δ(x) =
|f1(x) g1(x)|
|f2(x) g2(x)|
its differentiatin Δ'(x) =
|f1'(x) g1'(x)|
|f2(x) g2(x)|
Plus
|f1(x) g1(x)|
|f2'(x) g2'(x)|
This means to differentiate a determinant, we differentiate one row at a time, keeping others unchanged and then add them
If there are two rows R1 and R2
Δ'(x) =
|R1'|
|R2|
Plus
|R1|
|R2'|
|f1(x) g1(x)|
|f2(x) g2(x)|
its differentiatin Δ'(x) =
|f1'(x) g1'(x)|
|f2(x) g2(x)|
Plus
|f1(x) g1(x)|
|f2'(x) g2'(x)|
This means to differentiate a determinant, we differentiate one row at a time, keeping others unchanged and then add them
If there are two rows R1 and R2
Δ'(x) =
|R1'|
|R2|
Plus
|R1|
|R2'|
Friday, November 7, 2008
IIT JEE Mathematics Study Guide 12. Determinants
12.1 Definition
12.2 Singular matrix
12.3 Minors and cofactors
12.4 Properties of determinants
12.5 Evaluation of determinants
12.6 Evaluation of determinants by using factor theorem
12.7 Product of determinants
12.8 Differentiation of determinants
12.9 Applications of determinants to coordinate geometry
12.10 Applications of determinants in solving a system of linear equations
Study Plan
Day 1
12.1 Definition
There are three examples in this section
12.2 Singular matrix
12.3 Minors and cofactors
Day 2
12.4 properties of determinants
12.5 Evaluation of Determinants
Day 3
Do all the 19 example problems
Day 4
Study sections
12.6 Evaluation of Determinants using Factor Theorem
2 examples
12.7 Product of determinants
5 examples
Day 5
Do illustrative obj type examples 1 to 20
Day 6
12.8 Differentiation of determinants
12.9 Applications of determinants to coordinate geometry
Day 7
12.10 Applications of determinants in solving a system of linear equations
Day 8
I.O.T.E. 21 to 28
Objective Type Exercises 1 to 10
Day 9
O.T.E. 11 to 30
Day 10
O.T.E. 331 to 50
Day 11
O.T.E. 51 to 72
Day 11
Fill in the blansk 1 to 24
Day 12
True or false 6
Day 13 to 30
Revision Concepts and test paper problems
12.2 Singular matrix
12.3 Minors and cofactors
12.4 Properties of determinants
12.5 Evaluation of determinants
12.6 Evaluation of determinants by using factor theorem
12.7 Product of determinants
12.8 Differentiation of determinants
12.9 Applications of determinants to coordinate geometry
12.10 Applications of determinants in solving a system of linear equations
Study Plan
Day 1
12.1 Definition
There are three examples in this section
12.2 Singular matrix
12.3 Minors and cofactors
Day 2
12.4 properties of determinants
12.5 Evaluation of Determinants
Day 3
Do all the 19 example problems
Day 4
Study sections
12.6 Evaluation of Determinants using Factor Theorem
2 examples
12.7 Product of determinants
5 examples
Day 5
Do illustrative obj type examples 1 to 20
Day 6
12.8 Differentiation of determinants
12.9 Applications of determinants to coordinate geometry
Day 7
12.10 Applications of determinants in solving a system of linear equations
Day 8
I.O.T.E. 21 to 28
Objective Type Exercises 1 to 10
Day 9
O.T.E. 11 to 30
Day 10
O.T.E. 331 to 50
Day 11
O.T.E. 51 to 72
Day 11
Fill in the blansk 1 to 24
Day 12
True or false 6
Day 13 to 30
Revision Concepts and test paper problems
Saturday, June 7, 2008
Ch. 12 Determinants - 2
12.9
Determinants coordinate geometry
The area of a triangle having vertices at (x1,y1), (x2,y2) an d(x3,y3) us given by
Δ = determinants of
|x1 y1 1|
|x2 y2 1| *(1/2)
|x3 y3 1|
Condition of collinearity of three points:
Let the three points be (x1,y1), (x2,y2) and (x3,y3). The area of triange formed by these three points is zero. Hence the determinant
|x1 y1 1|
|x2 y2 1| will be zero
|x3 y3 1|
Equation of a line passing through two points:
Let the two points be (x1,y1), (x2,y2). If (x,y) is a point on the line passing through these two points, then (x1,y1), (x2,y2) and (x,y) are collinear. Hence the determinant
|x y 1|
|x1 y1 1|
|x2 y2 1| will be zero
Determinants coordinate geometry
The area of a triangle having vertices at (x1,y1), (x2,y2) an d(x3,y3) us given by
Δ = determinants of
|x1 y1 1|
|x2 y2 1| *(1/2)
|x3 y3 1|
Condition of collinearity of three points:
Let the three points be (x1,y1), (x2,y2) and (x3,y3). The area of triange formed by these three points is zero. Hence the determinant
|x1 y1 1|
|x2 y2 1| will be zero
|x3 y3 1|
Equation of a line passing through two points:
Let the two points be (x1,y1), (x2,y2). If (x,y) is a point on the line passing through these two points, then (x1,y1), (x2,y2) and (x,y) are collinear. Hence the determinant
|x y 1|
|x1 y1 1|
|x2 y2 1| will be zero
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