More Properties of Determinant
Solution: Determinant of this matrix = 3( 0 x 0-(-5 x-1)) –(-1)( 0 x 0 –(-1 x 3)) +(-2)( 0 x-5- 0 x3)
= 3(-5) + 3 + 0 =-12
Rule: If A = kB where A & B are square matrices of order n, then | A | = k n | B |, where n = 1, 2, 3
Numerical:
Solution:
det( 2A) = | 2A | = 2 x 4-4 x 8 = 8 – 32 =-24 det( a) = | A | = 2 x 1 – 4 x 2 =-6
It can be seen that | 2A | = 4 | A |
More Properties of Determinant
o
o
o
o
o
o
The value of determinant remains unchanged if its rows and columns are interchanged.
If any two rows( or columns) of a determinant are interchanged, then sign of determinant changes
If any two rows( or columns) of a determinant are identical( all corresponding elements are same), then value of determinant is zero.
If each element of a row( or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k.
If some or all elements of a row or column of a determinant are expressed as sum of two( or more) terms, then the determinant can be expressed as sum of two( or more) determinants
If, to each element of any row or column of a determinant, the equi multiples of corresponding elements of other row( or column) are added, then value of determinant remains the same, i. e., the value of determinant remain same if we apply the operation Ri → Ri + kRj or Ci → Ci + k Cj.