Determinants in linear algebra

In summary, in calculating the determinant of a matrix, it is important to remember that swapping two rows or columns changes the sign of the determinant. This applies to both rows and columns, even though it may not have been explicitly stated in class notes.
  • #1
Square1
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Homework Statement



i can't for the life of me find out where the negative comes from!

if you have the following the following determinant to calculate

det
(0) (1) (3) (4)
(2) (3) (1) (1)
(4) (-1) (-1) (2)
(6) (4) (-1) (1)and the next step shows... it equals (note negative sign)= -det
(2) (3) (1) (1)
(0) (1) (3) (4)
(4) (-1) (-1) (2)
(6) (4) (-1) (1)
how did they get the negative! which rule is this! The answer is -32 which. If i do it my way without adding the negative in at the first step, i get 32 so the rest of the steps i do are right.

Thanks!
 
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  • #2
It's a general property of determinants that if you swap two rows or cols, it changes the sign of the determinant.
 
  • #3
OMG ROWS TOO?? god it doesn't tell us that in our notes! only columns! thank you!
 

FAQ: Determinants in linear algebra

1. What are determinants in linear algebra?

Determinants in linear algebra are mathematical values that are used to determine properties of matrices, such as whether they are invertible or not. They are calculated using a specific formula and can provide important information about the matrix, such as its volume or area.

2. How do you calculate determinants?

The determinant of a matrix is calculated by multiplying the elements in a specific pattern and then subtracting the products of the other elements. The specific pattern is determined by the size of the matrix, and there are different methods for calculating determinants of larger matrices.

3. What is the significance of determinants in linear algebra?

Determinants have several important applications in linear algebra, such as determining whether a matrix is invertible or not, finding the area or volume of a matrix, and solving systems of linear equations. They also play a crucial role in other areas of mathematics, such as calculus and differential equations.

4. Can determinants be negative?

Yes, determinants can be negative. The sign of a determinant depends on the number of row swaps needed to reduce the matrix to its reduced row echelon form. If an odd number of row swaps is needed, the determinant will be negative, and if an even number is needed, the determinant will be positive.

5. How are determinants used in solving systems of linear equations?

Determinants can be used to determine whether a system of linear equations has a unique solution, infinite solutions, or no solutions. If the determinant of the coefficient matrix is equal to zero, the system has either infinite solutions or no solutions, depending on the consistency of the equations. If the determinant is non-zero, then the system has a unique solution.

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