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

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Discussion Overview

The discussion revolves around finding the determinant of the inverse of a 3x3 matrix B, given the relationship between matrices A, B, and C, along with specific determinant values. The scope includes mathematical reasoning and properties of determinants in linear algebra.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant states that since \( ABC = I \), it follows that \( |ABC| = |I| \) leading to the equation \( |A||B||C| = 1 \).
  • Another participant proposes that if \( |3A| = 81 \), then \( |A| \) can be determined as 27, but questions how to find \( |C| \) given that the information provided is about \( C^T \).
  • It is noted that properties of determinants imply \( |C^T| = |C| \) and \( |B^{-1}| = \frac{1}{|B|} \), but the exact value of \( |B| \) remains undetermined.
  • There is a reiteration of the calculation \( |3A| = 3^3 |A| = 27 |A| = 81 \), suggesting a connection to the determinant of A.

Areas of Agreement / Disagreement

Participants express uncertainty about determining the value of \( |C| \) and \( |B| \), indicating that multiple views on how to approach the problem exist, and the discussion remains unresolved.

Contextual Notes

Limitations include the lack of explicit values for \( |B| \) and \( |C| \), as well as the dependence on the properties of determinants without a complete resolution of the mathematical steps involved.

mahmoud shaaban
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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.
 
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mahmoud shaaban said:
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)
 
I know that A = 27 ,but how can i know what C = ?? if the given is C^T
 
mahmoud shaaban said:
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)
 

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