Properties Of Determinants

Let

P%20%3D%5Cbegin%7Bbmatrix%7Da%26b%5C%5C%20c%26d%5Cend%7Bbmatrix%7D

Given,

%5Ctherefore%5Cleft%5B1%200%5Cright%5DP%20%3D%20%5Cfrac%7B-1%7D%7B%5Csqrt%7B2%7D%7D%20%5Cleft%5B1%201%5Cright%5D

%5Cleft%5B10%5Cright%5D%5Cbegin%7Bbmatrix%7Da%26b%5C%5C%20c%26d%5Cend%7Bbmatrix%7D%20%3D%20%5Cleft%5B-%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%20%5Cfrac%7B-1%7D%7B%5Csqrt%7B2%7D%7D%5Cright%5D

%5Cleft%5Bab%5Cright%5D%20%3D%20%5Cleft%5B%5Cfrac%7B-1%7D%7B%5Csqrt%7B2%7D%7D%20%5Cfrac%7B-1%7D%7B%5Csqrt%7B2%7D%7D%5Cright%5D

and

%5Cleft%5B%2001%5Cright%5DP%20%3D%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%5Cleft%5B-1%201%5Cright%5D

%5Cleft%5B0%201%5Cright%5D%5Cbegin%7Bbmatrix%7Da%26b%5C%5C%20c%26d%5Cend%7Bbmatrix%7D%20%3D%20%5Cleft%5B%20-%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%5Cright%5D

%5Cleft%5Bcd%5Cright%5D%20%3D%20%5Cleft%5B%20-%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%5Cright%5D

%5Ctherefore%20a%20%3D%20%5Cfrac%7B-1%7D%7B%5Csqrt%7B2%7D%7D%2C%20b%3D%20%5Cfrac%7B-1%7D%7B%5Csqrt%7B2%7D%7D%2C%20c%20%3D%20%5Cfrac%7B-1%7D%7B%5Csqrt%7B2%7D%7D%2C%20d%20%3D%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D

P%20%3D%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%5Cbegin%7Bbmatrix%7D-1%26-1%5C%5C%20-1%261%5Cend%7Bbmatrix%7D

P%5E%7B2%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%5Cbegin%7Bbmatrix%7D-1%26-1%5C%5C%20-1%261%5Cend%7Bbmatrix%7D%5Cbegin%7Bbmatrix%7D-1%26-1%5C%5C%20-1%261%5Cend%7Bbmatrix%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D%5Cbegin%7Bbmatrix%7D2%260%5C%5C%200%262%5Cend%7Bbmatrix%7D

P%5E%7B2%7D%20%3D%20I

P%5E%7B4%7D%20%3D%20I%20%3D%20P%5E%7B6%7D%20%3D%20P%5E%7B8%7D

P%5E%7B8%7D%20%2BP%5E%7B6%7D%20%2BP%5E%7B4%7D%20-P%5E%7B4%7D%20-P%5E%7B2%7D

%3D%20I%20%2B%20I%2B%20I%20-I%20%3D%202I

Chapter Questions
2 MCQs