How does subtracting a row from another affect the determinant?

In summary, the conversation is about reducing a matrix to row echelon form to find the determinant. The original matrix has a determinant of 0.8, but after subtracting the 4th row from the 3rd row, the determinant changes to -0.8. The person is confused about this change and asks for clarification. The conversation ends with the correction that the bottom right entry should be +0.8 instead of -0.8.
  • #1
fleeceman10
14
0

Homework Statement


I think I have broken maths. I am reducing a matrix to row echelon form to find the determinant. The matrix I will show is nearly in the desired form


Homework Equations


1 -3 -2 1
0 1 2 -1
0 0 1 -0.8
0 0 1 0


The Attempt at a Solution


As it stands, the determinant is 0.8. However subtracting the 4th row from the 3rd row changes the determinant to -0.8. I thought adding multiples of one row to another left the determinant unchanged. So why hasn't it? Thanks for answers.
 
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  • #2
Doing that leaves the determinant at +0.8.

Det(The matrix)=det(The matrix without the first row and first column)=det(The matrix without the first and second row and the first and second column).

I arrive at that from expansion along the first column twice.

So you arrive at det(The matrix)=(1)(0)-(1)(-0.8)=0.8

Subtracting the row like you said just zaps that first 1 to 0 making it (0)(0)-(1)(-0.8)=0.8.

Check your calculation again.
 
  • #3
Pagan Harpoon said:
Doing that leaves the determinant at +0.8.

Det(The matrix)=det(The matrix without the first row and first column)=det(The matrix without the first and second row and the first and second column).

I arrive at that from expansion along the first column twice.

So you arrive at det(The matrix)=(1)(0)-(1)(-0.8)=0.8

Subtracting the row like you said just zaps that first 1 to 0 making it (0)(0)-(1)(-0.8)=0.8.

Check your calculation again.

Am I correct in thinking that the matrix obtained doing this calculation (3rd row - 4th row) is this
1 | -3 | -2 | 1
0 | 1 | 2 | -1
0 | 0 | 1 | -0.8
0 | 0 | 0 | -0.8

Since this is a triangular matrix, the determinant is the product of the entries on the leading diagonal, -0.8. Where have I gone wrong?
 
  • #4
You originally said subtracting the 4th from the 3rd, which would make the bottom corner

0 -0.8
1 0

That is how I would usually interpret subtracting the 4th from the 3rd row, anyway.

So you mean what I would call subtracting the 3rd from the 4th row. You have done the calculation incorrectly, the bottom right entry should be +0.8 because it is 0-(-0.8).
 
  • #5


It is important to note that row operations do not change the value of the determinant, but they may change the sign of the determinant. In this case, subtracting the 4th row from the 3rd row changes the determinant from 0.8 to -0.8 because it is equivalent to multiplying the determinant by -1. This is because this operation essentially swaps the 3rd and 4th rows, which changes the order of the rows and thus changes the sign of the determinant. So, your calculation is correct and the determinant is indeed -0.8. Keep up the good work!
 

What is a determinant in linear algebra?

A determinant is a numerical value that is associated with a square matrix. It is calculated by a specific formula and provides important information about the properties of the matrix, such as whether it is invertible or singular.

How is the determinant of a matrix calculated?

The determinant of a matrix is calculated by summing the products of certain elements in the matrix according to a specific pattern. This pattern is known as the "cofactor expansion" method, and it involves repeatedly removing a row and column from the matrix and calculating the determinant of the resulting smaller matrix.

What is the significance of the determinant in linear algebra?

The determinant is a useful tool in linear algebra because it provides important information about the properties of a matrix. For example, if the determinant is non-zero, the matrix is invertible and has a unique solution to the system of equations it represents. If the determinant is zero, the matrix is singular and has either no solution or infinitely many solutions.

Can the determinant of a matrix be negative?

Yes, the determinant of a matrix can be negative. The determinant is simply a numerical value and can be positive, negative, or zero. In fact, the sign of the determinant can provide information about the orientation of the vector space spanned by the columns or rows of the matrix.

How is the determinant used in solving systems of linear equations?

The determinant is used in solving systems of linear equations by determining whether the system has a unique solution, no solution, or infinitely many solutions. If the determinant is non-zero, the system has a unique solution. If the determinant is zero, the system either has no solution or infinitely many solutions, depending on the specific equations. The determinant can also be used to find the inverse of a matrix, which is useful in solving systems of equations.

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