# Thread: These Determinants are driving me crazy! (Matrices)

1. ## These Determinants are driving me crazy! (Matrices)

Hey guys,

Firstly, Happy Thanks Giving for all of you who are Canadian! I've got a few problems with determinants which I can't solve! I'm hoping any of you can help me:

I know the answer to the last problem of the first question is -23. Can anyone show me the rest? I thank you in advanced!

2. Originally Posted by JoeyCC
Hey guys,

Firstly, Happy Thanks Giving for all of you who are Canadian! I've got a few problems with determinants which I can't solve! I'm hoping any of you can help me:

I know the answer to the last problem of the first question is -23. Can anyone show me the rest? I thank you in advanced!
for (i):

$\displaystyle \det(\text{adj}(A^T))=\det((\text{adj}(A))^T)=\det (\text{adj}(A))=(\det (A))^2=25.$ in general for an $\displaystyle n \times n$ matrix $\displaystyle A$ we have $\displaystyle \det(\text{adj}(A))=(\det(A))^{n-1}.$

$\displaystyle \det(\text{adj}(A^{-1}))=\frac{1}{(\det(A))^2}=\frac{1}{25}.$

$\displaystyle \det(\text{adj}(8A))=8^6(\det(A))^2=8^6 \times 25=6553600.$

for the first part of (ii) see my answer to the first part of (i). for the second part do a couple of elementary row opertaions to show that:

$\displaystyle \det \begin{bmatrix} v_1 \\ 7v_2 + 2v_4 \\ v_3 \\ 4v_2 + 5v_4 \end{bmatrix} = \det \begin{bmatrix} v_1 \\ 7v_2 \\ v_3 \\ \frac{27}{7}v_4 \end{bmatrix}=27 \det \begin{bmatrix} v_1 \\ v_2 \\ v_3 \\ v_4 \end{bmatrix}= 27 \times 6 = 162.$