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PROPERTY:1. The determinant of a square zero matrix is zero. : PROPERTY:1. The determinant of a square zero matrix is zero.
PROPERTY: 2. The determinant of a matrix whose all entries in any row or column are zeroes is zero. : PROPERTY: 2. The determinant of a matrix whose all entries in any row or column are zeroes is zero.
PROPERTY:3. The determinant of a triangular matrix is obtained by the product of elements in the main diagonal. : PROPERTY:3. The determinant of a triangular matrix is obtained by the product of elements in the main diagonal.
PROPERTY:4. The det. of a diagonal matrix is equal to product of its diagonal elements. : PROPERTY:4. The det. of a diagonal matrix is equal to product of its diagonal elements.
PROPERTY:5. If rows and columns of a determinant are interchanged the value of the determinant remains unaltered : PROPERTY:5. If rows and columns of a determinant are interchanged the value of the determinant remains unaltered
PROPERTY:6. If we interchange any two parallel lines of a determinant the value of the determinant changes in sign.(Magnitude remains the same ) : PROPERTY:6. If we interchange any two parallel lines of a determinant the value of the determinant changes in sign.(Magnitude remains the same )
PROPERTY:7. If two parallel lines of a determinant are identical, its value is zero. Cor. If two parallel lines of a determinant have proportional elements, the value of determinant is also zero : PROPERTY:7. If two parallel lines of a determinant are identical, its value is zero. Cor. If two parallel lines of a determinant have proportional elements, the value of determinant is also zero
PROPERTY 8. If the elements of a line (row or column) are multiplied by a constant ‘k’ , then the whole determinant gets multiplied by that constant (k) Cor. : We can pull out a common factor from the elements of any row or column : PROPERTY 8. If the elements of a line (row or column) are multiplied by a constant ‘k’ , then the whole determinant gets multiplied by that constant (k) Cor. : We can pull out a common factor from the elements of any row or column
PROPERTY 9. If each element of a row/ column consists of two or more terms, then the determinant can be expressed as the sum of as many determinants. Cor. If there are ‘m’ terms each in one column and ‘p’ terms each in another column ,then number of determinants that can be formed is m.p : PROPERTY 9. If each element of a row/ column consists of two or more terms, then the determinant can be expressed as the sum of as many determinants. Cor. If there are ‘m’ terms each in one column and ‘p’ terms each in another column ,then number of determinants that can be formed is m.p
PROPERTY 10. If we add the equimultiples of the elements one line to the corresponding elements of another parallel line, then value of the new determinant remains the same. : PROPERTY 10. If we add the equimultiples of the elements one line to the corresponding elements of another parallel line, then value of the new determinant remains the same. This operation is denoted by
PROPERTY 11.If we multiply the elements of a line of a determinant by their own co-factors, then their sum is equal to value of the determinant .There are such 6 relations in a third order determinant. : PROPERTY 11.If we multiply the elements of a line of a determinant by their own co-factors, then their sum is equal to value of the determinant .There are such 6 relations in a third order determinant. then
PROPERTY 12. If we multiply the elements of a row/column by the corresponding co-factors of another row/column, then their sum is zero. This is know as ‘zero relation’ . : PROPERTY 12. If we multiply the elements of a row/column by the corresponding co-factors of another row/column, then their sum is zero. This is know as ‘zero relation’ . For example
PROPERTY 13: The product of two determinants of the same order can also be written in the form of a determinants & is obtained by any one of the following rules: (i) (Row x Column) multiplication ( ii) (Row x Row ) multiplication(iii) (Column x Row) multiplication ( iv) (Column x Column) multiplication : PROPERTY 13: The product of two determinants of the same order can also be written in the form of a determinants & is obtained by any one of the following rules: (i) (Row x Column) multiplication ( ii) (Row x Row ) multiplication(iii) (Column x Row) multiplication ( iv) (Column x Column) multiplication For example
PROPERTY 14. If “A” is any square matrix & ‘C’ is the Matrix formed by its co-factors, then and Where ‘n’ is the order of determinants of A : PROPERTY 14. If “A” is any square matrix & ‘C’ is the Matrix formed by its co-factors, then and Where ‘n’ is the order of determinants of A For example
PROPERTY 15: If ‘A & ‘B’ are two square matrices of same order then Cor. : PROPERTY 15: If ‘A & ‘B’ are two square matrices of same order then Cor. Inverse of a square matrix exists if its determinants is not zero PROPERTY 15: If ‘A & ‘B’ are two square matrices of same order then Cor.
PROPERTY 16:If represent the columns of a determinant and are functions of ‘x’, then the derivative of the determinant is given by: : PROPERTY 16:If represent the columns of a determinant and are functions of ‘x’, then the derivative of the determinant is given by: For example
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