Finding Solutions for a Matrix with Variables: Row Reduction Required?

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SUMMARY

The discussion focuses on determining the values of the variable x in the matrix A = (1 -2 3 1; 2 x 6 6; -1 -3 x-3 0) that result in no solutions, one solution, or infinitely many solutions. Key concepts include the geometric interpretation of solutions in relation to row reduction and the row reduced echelon form (RREF). Participants emphasize the importance of identifying values of x that lead to undefined conditions, such as denominators equating to zero during the algebraic manipulation. A clear distinction is made that a matrix itself does not have a solution; rather, it is the matrix equation that yields solutions based on the values of x.

PREREQUISITES
  • Understanding of matrix algebra and operations
  • Familiarity with row reduction techniques and row reduced echelon form (RREF)
  • Basic knowledge of algebraic manipulation involving variables
  • Geometric interpretation of linear equations and their solutions
NEXT STEPS
  • Study the process of row reduction to obtain the row reduced echelon form (RREF) of matrices
  • Learn about the conditions for a system of linear equations to have no solutions, one solution, or infinitely many solutions
  • Explore the implications of variable denominators in algebraic expressions
  • Investigate the geometric interpretation of linear equations in three-dimensional space
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Students and educators in linear algebra, mathematicians analyzing systems of equations, and anyone interested in the properties of matrices and their solutions.

rhuelu
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here's a tough one...

find the values of x for which the matric has no, one, and inf many solutions

A= (1 -2 3 1; 2 x 6 6; -1 -3 x-3 0)
 
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What do all those mean to you? I.e. what properties does a matrix that has no solutions have? What about one solution? What about infinitely many? You can think of these geometrically, what does it mean (geometricall) to have no solutions? one? infinitely many?
 
What have you done? If you want, as your title implies the the "row reduced echelon form", then go ahead and do the row reduction! The fact that "x" is a variable only means you have to use a little algebra rather than just arithmetic. That probably will involve fractions that have some function of x as the denominator. Obviously, values of x that make the denominator 0 will give difficulties.

I do feel compelled to point out that a matrix doesn't have a "solution". If you are asking for which values of x does the matrix equation has one, no, or infinitely many solutions, you will have to have an equation!
 

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