Finding Matrix B from given info

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SUMMARY

The discussion centers on solving for matrix B in the equation (I + 3B)^-1 = [5 2; 4 2]. The user attempted to isolate I by multiplying both sides by (I + 3B) and substituting B with a general 2x2 matrix [a b; c d]. Despite following the algebraic steps, the user did not arrive at the correct matrix B. A suggestion was made to calculate the inverse of the 2x2 matrix on the right-hand side to simplify the process.

PREREQUISITES
  • Understanding of matrix algebra, specifically matrix inversion.
  • Familiarity with 2x2 matrices and their properties.
  • Knowledge of the identity matrix and its role in matrix equations.
  • Ability to manipulate algebraic expressions involving matrices.
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  • Learn how to calculate the inverse of a 2x2 matrix.
  • Study the properties of the identity matrix in matrix operations.
  • Explore techniques for solving matrix equations involving unknown matrices.
  • Review elementary matrix algebra to strengthen foundational skills.
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Homework Statement


Use the given info to find matrix B

Homework Equations


(I + 3B)^-1 = [5 2; 4 2]

to make more clear:

inv(I + 3B) = this 2x2 matrix: top row = 5 2, bottom row = 4 2

The Attempt at a Solution


I tried multiplying both sides of the eqn by I + 3B to get I = [5 2; 4 2](I + 3B)
Then tried to multiply the two quantities (using B = [a b; c d])
Tried to set that product equal to the identity matrix and solve for the unspecified constants, a b c d.
After that I plugged the values into the the product found in the previous step
I then should have acquired the correct matrix B, but alas, it wasn't even close.
If I am overlooking some elementary matrix 'algebra', or any hints at all would be greatly appreciated.
Thank you!
 
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Why not just calculate the matrix inverse of the 2x2 matrix on the RHS?
 

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