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
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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 .