Question

You are given that A and B are two square matrices of the same order, such that

$\mathrm{det}\left(A^{-1}B\right)=20\quad \mathrm{and} \quad \mathrm{det} \left(B\right)=5.$

Which of the following is true?

1. $\mathrm{det}(A)=-15\quad \text{and} \quad \mathrm{adj}(A)=-\frac{1}{15}$
2. $\mathrm{det}(A)=4\quad \text{and} \quad\mathrm{adj}(A)=4A^{-1}$
3. $\mathrm{det}(A)=\frac{1}{4}\quad \text{and} \quad \mathrm{adj}(A)=\frac{1}{4}A^{-1}$
4. $\mathrm{det}(A)=25\quad \text{and} \quad \mathrm{adj}(A)=25$
5. $\mathrm{det}(A)=15\quad \text{and} \quad \mathrm{adj}(A)=15 A^{-1}$

.

Answer 1

$\det(A^{-1}B)=\det{A^{-1}}\det{B}\\=\dfrac{\det{B}}{\det{A}}\\ 20=\dfrac{5}{\det{A}}\\ \det{A}=\dfrac{1}{4}\\ \text{Inverse formula }A^{-1}=\dfrac{\text{adj(A)}}{\det(A)}\\ \text{adj(A)}=(\det{A})A^{-1}=\dfrac{1}{4}A^{-1}$

True statement is 3