Determinant of a matrix using reduced echelon form

In summary, the determinant of the given matrix can be found by reducing it to upper triangular form, where the determinant is equal to the product of the entries on the diagonal. Since the second row is a linear combination of the other three, the determinant will be 0. This also applies to any matrix containing a row vector consisting of all zeros. This follows from the definition and properties of the determinant.
  • #1
TrippingBilly
27
0
Problem statement: Find the determinant of the following matrix by row reduction to echelon form.
|1 3 3 -4 |
|0 1 2 -5 |
|2 5 4 -3 |
|-3 -7 -5 2 |

I reduced this matrix to
|1 3 3 -4 |
|0 1 2 -5 |
|0 -1 -2 5 |
|0 2 4 -10 |

If I reduce this further, the entries in row 3 and 4 become 0. Is this still considered triangular form, and therefore the determinant will be 0?
 
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  • #2
TrippingBilly said:
Problem statement: Find the determinant of the following matrix by row reduction to echelon form.
|1 3 3 -4 |
|0 1 2 -5 |
|2 5 4 -3 |
|-3 -7 -5 2 |

I reduced this matrix to
|1 3 3 -4 |
|0 1 2 -5 |
|0 -1 -2 5 |
|0 2 4 -10 |

If I reduce this further, the entries in row 3 and 4 become 0. Is this still considered triangular form, and therefore the determinant will be 0?

Correct. It was obvious from the beginning, since the third and second row are proportional (after your first "reduction").
 
Last edited:
  • #3
Yes and yes.
 
  • #4
Ok, I didn't reduce it by row-eschelon form, but you should get a determinant of zero.

Why? The second row is a linear combination of the other three.

Upper Triangular form just requires that the entries below the diagonal are all zero. If entries above (or on) the diagonal are zero, that's ok.

ZM
 
  • #5
If this is true, then will the determinant of any matrix containing a row vector consisting of all zeros will be zero?
 
  • #6
TrippingBilly said:
If this is true, then will the determinant of any matrix containing a row vector consisting of all zeros will be zero?

Yes, it will.
 
  • #7
Yes. Think of it this way:

The determinant function can be viewed as a machine that takes n n-dimensional vectors and spits out a number. However, if any of these vectors are linear combinations of any of the other vectors, then the determinant will be zero. A zero vector is a linear combination of every vector.

Another way to see the same thing is just to break the determinant into minors along the row (or column) of zeros. The determinant is then zero.

ZM
 
  • #8
The exact reason follows directly from the http://www.cs.ut.ee/~toomas_l/linalg/lin1/node14.html" of the determinant.
 
Last edited by a moderator:

1. What is the purpose of finding the determinant of a matrix using reduced echelon form?

Finding the determinant of a matrix using reduced echelon form allows us to efficiently solve systems of linear equations and determine whether a matrix is singular or nonsingular.

2. How do you find the determinant of a matrix using reduced echelon form?

To find the determinant using reduced echelon form, we first reduce the matrix to its reduced echelon form. Then, we multiply the elements on the main diagonal of the reduced matrix. This product is the determinant of the original matrix.

3. Can the determinant of a matrix be negative?

Yes, the determinant of a matrix can be negative. The sign of the determinant depends on the number of row exchanges made during the process of reducing the matrix to its reduced echelon form.

4. Is it necessary to use reduced echelon form to find the determinant of a matrix?

No, it is not necessary to use reduced echelon form to find the determinant of a matrix. The determinant can also be found by using other methods such as cofactor expansion or Gaussian elimination.

5. What is the significance of the determinant of a matrix?

The determinant of a matrix has many applications in mathematics, including solving systems of linear equations, finding the inverse of a matrix, and determining whether a matrix is singular or nonsingular. It also has applications in other fields such as physics and economics.

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