Finding B^-1 in 3x3 Matrices with Linear Algebra

[mahmoud shaaban]

if A and B are 3x3 matrices such that: ABC = I, |3A|=81 and |C^T|= 2 , how to find |B^-1|

I couldn't solve this because there is not much given.

 

[I like Serena]

if A and B are 3x3 matrices such that: ABC = I, |3A|=81 and |C^T|= 2 , how to find |B^-1|

I couldn't solve this because there is not much given.

Hi mahmoud shaaban!

We are given:
$$ABC=I \Rightarrow
|ABC|=|I| \Rightarrow
|A||B||C|=1
$$
Can we find $|A|$ and $|C|$? (Wondering)

 

[mahmoud shaaban]

I know that A = 27 ,but how can i know what C = ?? if the given is C^T

 

[I like Serena]

I know that A = 27 ,but how can i know what C = ?? if the given is C^T

Properties of the determinant are that:
$$|xA|=x^n|A| \\
|C^T|=|C| \\
|B^{-1}|=\frac{1}{|B|}$$

Oh, that also means that $|3A|=3^3|A|=27|A|=81$. (Thinking)