Test for compatibility of equations - Determinant |A b|

In summary, the conversation discusses the use of the determinant of an augmented matrix $|A\ b|$ as a test for the compatibility of a system of linear equations. The test determines if the system is solvable or not based on the number of equations and unknown variables. It is also mentioned that the determinant being zero indicates a dependent system. The conversation also clarifies the difference between "compatibility" and "combatibility" and the importance of independent equations.
  • #1
mathmari
Gold Member
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Hey! :eek:

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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  • #2
mathmari said:
Hey! :eek:

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 $A^+b$?
 
  • #3
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)
 
  • #4
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)
 
  • #5
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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Related to Test for compatibility of equations - Determinant |A b|

1. What is a determinant?

A determinant is a mathematical value that can be calculated from a square matrix. It is used to determine whether a system of equations has a unique solution, no solution, or infinitely many solutions.

2. Why is the determinant important in testing equation compatibility?

The determinant is important because it can tell us whether a system of equations has a unique solution or not. If the determinant is equal to zero, the system has either no solution or infinitely many solutions. If the determinant is non-zero, the system has a unique solution.

3. How is the determinant calculated?

The determinant is calculated by using a specific formula depending on the size of the matrix. For a 2x2 matrix, the determinant is calculated by multiplying the elements in the main diagonal and subtracting the product of the elements in the other diagonal. For larger matrices, the process involves finding the determinants of smaller submatrices.

4. Can the determinant be negative?

Yes, the determinant can be negative. The sign of the determinant does not affect its use in determining equation compatibility. It only indicates the orientation of the matrix.

5. What does a zero determinant indicate?

A zero determinant indicates that the system of equations has either no solution or infinitely many solutions. This means that the equations are not compatible and cannot be solved uniquely.

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