7 Let A and B be n n matrices Determine which one of the fo

7. Let A and B be n × n matrices. Determine which one of the following is TRUE. (a) If det A =2, then det A1 = 2 (b) If B is not invertible, then det AB = 0. (c) det (5An×n) = 5det A (d) det(AB) = det(A) + det(B) (e) none of these

Solution

Ans(a):

If det A =2, then det A1 = 2

FALSE.

We can easily verify that by taking any 2x2 matrix having elements a,b,c,d.

Determinante will be (ad-bc) and hence determinant of inverse is 1/(ad-bc) so for given situation we get 1/2 not -2

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Ans(b):

Given B is not invertible means det(B)=0

we know det(AB)=det(A)*det(B)=det(A)*0=0

Hence it is TRUE

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Ans(C):

FALSE

Because 5Anxn mens each row is multiplied by 5 so in determinant we will get 5 for each row

Hence det (5An×n) = 5^n(det A)

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Ans(D):

FALSE

because det(AB)=det(A)*det(B)

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Ans(E):

FALSE

as we already got some true results :)