GO Classes DPP | GATE CS | Linear Algebra | Properties of Determinant

Answer 1

The statement that is not always true is $\boxed{\text{C}}$

because determinant is not additive in general, so $\det(A+kI)\neq \det(A)+k^n$ in general.

For example, take $A=I_2$ and $k=1$.

Then $\det(A+kI)=\det(2I_2)=4$, but $\det(A)+k^n=1+1=2$.

So, the equality fails.
 

The other statements are true:

A. $\det(AB^{-1}A^T)=\det(A)\det(B^{-1})\det(A^T)=\det(A)\cdot \dfrac{1}{\det(B)} \cdot \det(A)=\dfrac{(\det A)^2}{\det B}$

B. For an invertible matrix $A$ of order $n$, $\det(\operatorname{adj}(A))=(\det A)^{n-1}$

D. $\det(A^{-1}B^T)=\det(A^{-1})\det(B^T)=\dfrac{1}{\det A}\det B=\dfrac{\det B}{\det A}$

Hence, the correct answer is $\boxed{\text{C}}$.

Comments

for additive  it shold be  inequaltiy sign betwen them  .

Answer 2

The determinant is multiplicative, not additive: det(AB) = det(A)det(B), but det(A+B) generally does not equal det(A) + det(B).

 

Answer 3

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