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# Advanced Math/Row reducing a matrix-linear equations solutions

Question
Hi Randy,

Sorry for the long winded question

I know that when we are finding solutions for a linear equation there are usually three types of solution solutions. infinitely many solutions, one solution, and no solution. I am wondering if I were to use row reduction and matrices to solve linear equations how would I know when there is no solution or many solutions?
Would one row become 0,0,0]2 where the identity matrix can't be formed with row reduction and for example above there would be 0=2. Or is there a short cut where I could find the determinant of a 3x3 matrix which represent a x,y,z linear system and if the determinant = 0 there would be no solution. But my problem is can I find the determinant and use it to determine if a linear system is infinitely many solutions?

Many Many Thanks,
Sam

You need to specify whether the equations are homogeneous or not, that is

Ax = 0   homogenous

Ax = b  inhomogeneous.

For the homogenous case, the only way to have a non-zero solution (where x is a vector, x = (x1, x2, ..., xn), with at least one of its components non-zero), is if detA = 0. This can be seen by letting detA ≠ 0, in which case A has an inverse and x = [A^-1]0 = 0, so that x is all zeroes.

In a similar way, for the inhomogeneous case, we must have detA ≠ 0 for there to be a solution. The solution obtained is unique. On the other hand, if detA = 0, then the solution to the inhomogeneous equation is not unique.

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#### randy patton

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college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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