Test for compatibility of equations - Determinant |A b|

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SUMMARY

The discussion centers on the compatibility of linear equations represented by the system $Ax=b$, where the matrix $A$ has full column rank and the number of equations exceeds the number of unknowns. Participants clarify that the determinant $|A \ b|$ is used to assess the solvability of the system, specifically addressing the relationship between the augmented matrix $(A \mid b)$ and the conditions for compatibility. It is established that if the equations are independent, no solution exists, while dependent equations yield a determinant of zero, indicating potential solutions.

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mathmari
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Hey! :o

Let $Ax=b$ be a system of linear equations, where the number of equations is by one larger than the number of unknown variables, so the matrix $A$ is of full column rank.

Why can the test for combatibility of equations use the criterion of the determinant $|A \ b|$ ? (Wondering)
 
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mathmari said:
Hey! :o

Let $Ax=b$ be a system of linear equations, where the number of equations is by one larger than the number of unknown variables, so the matrix $A$ is of full column rank.

Why can the test for combatibility of equations use the criterion of the determinant $|A \ b|$ ?

Hey mathmari!

What is this "test for compatibility of equations"? (Wondering)

How is $|A \ b|$ a determinant? It's not a square matrix is it? (Wondering)
Did you perhaps mean the Matlab notation [M]A \ b[/M], which means $A^+b$?
 
Klaas van Aarsen said:
What is this "test for compatibility of equations"? (Wondering)

The test if the system is solvable.
Klaas van Aarsen said:
How is $|A \ b|$ a determinant? It's not a square matrix is it? (Wondering)
Did you perhaps mean the Matlab notation [M]A \ b[/M], which means $A^+b$?

But isn't the augmented matrix $(A\mid b)$ a square matrix, since the number of equations is by one larger than the number of unknown variables and that means that $A$ is a $n\times (n-1)$ matrix, or not? (Wondering)
 
mathmari said:
The test if the system is solvable.

But isn't the augmented matrix $(A\mid b)$ a square matrix, since the number of equations is by one larger than the number of unknown variables and that means that $A$ is a $n\times (n-1)$ matrix, or not?

Ah okay.

If the system has a solution, then $b$ must be in the column space of $A$ yes? (Wondering)
That is, we can write $b$ as a linear combination of the column vectors in $A$.

Doesn't that imply that the determinant of the augmented matrix $(A\mid b)$ is zero? (Thinking)
 
I suspect you mean "compatibility", not "combatibiity"!

No one wants equations to fight!

If the number of equations is greater than the number of unknowns and the equations are "independent" there is no solution. There may be a solution if the equations are "dependent" which means the determinant must be 0.
 
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