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 det. of a matrix whose all entries in any row or column are zeroes is zero. : PROPERTY: 2. The det. 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