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

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SUMMARY

The discussion focuses on finding the inverse of matrix B (|B^-1|) given the equations involving three 3x3 matrices A, B, and C, where ABC = I, |3A| = 81, and |C^T| = 2. It is established that |A| equals 27, derived from the property |3A| = 3^3|A|. The relationship |ABC| = |I| leads to the conclusion that |A||B||C| = 1, allowing for the calculation of |B| once |C| is determined. The determinant properties discussed are crucial for solving for |B^-1|.

PREREQUISITES
  • Understanding of 3x3 matrix properties
  • Familiarity with determinants and their calculations
  • Knowledge of matrix inverses and their relationships
  • Basic concepts of linear algebra
NEXT STEPS
  • Study the properties of determinants in linear algebra
  • Learn how to compute the inverse of a matrix
  • Explore the implications of matrix multiplication on determinants
  • Investigate the relationship between a matrix and its transpose
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as anyone involved in computational mathematics or engineering applications requiring matrix operations.

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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