How is a determinant found using expansion by minors?
The only way I was taught to do it was by using diagonals.
check this page
http://mathworld.wolfram.com/DeterminantExpansionbyMinors.html
It is presented better than we could do on this format
upper left * det of the rest without that row and column
- second down on left times det of everything without that row and column
+ third down on left times det of everything without that row and column
etc down that left column alternating signs until you get to the bottom left.
In the trivial case of
a b
c d
it would be
a (d)
-c(b)
which you knew already
thanks
now for the slightly less trivial
a b c
d e f
g h i
a times det
e f
h i
- d times det
b c
h i
+ g times det
b c
e f
which is
a (e i - h f)
-d(b i - h c)
+g(b f - e c )
which is
a e i - a h f - d b i + d h c + g b f - g e c
which is the same as you get using the diagonal method
Expansion by minors is another method to find the determinant of a matrix. It involves recursively computing determinants of smaller matrices known as minors.
Here's an explanation of how to find the determinant using expansion by minors:
1. Choose any row or column of the matrix that you want to expand the determinant along. Let's say you choose the first row for reference.
2. For each element in the chosen row (or column), compute the determinant of the remaining matrix after removing the row and column containing the element. These smaller matrices are called minors.
3. Multiply each element in the chosen row (or column) by the determinant of its corresponding minor. These products are called cofactors.
4. Add up the products from Step 3 to obtain the determinant.
Let's illustrate this process with a specific example. Consider the following 3x3 matrix:
| a b c |
A = | d e f |
| g h i |
Let's find the determinant of matrix A:
1. Choose the first row for expansion. So, we'll focus on elements 'a', 'b', and 'c'.
2. Compute the determinants of three 2x2 matrices, which are minors of matrix A:
- For element 'a':
- Remove the first row and first column, we obtain the minor matrix:
| e f |
| h i |
- Compute its determinant: det(| e f |) = (e * i) - (f * h)
(| h i |)
- For element 'b':
- Remove the first row and second column, we obtain the minor matrix:
| d f |
| g i |
- Compute its determinant: det(| d f |) = (d * i) - (f * g)
(| g i |)
- For element 'c':
- Remove the first row and third column, we obtain the minor matrix:
| d e |
| g h |
- Compute its determinant: det(| d e |) = (d * h) - (e * g)
(| g h |)
3. Multiply each element in the first row ('a', 'b', and 'c') by its corresponding cofactor (determinant of the respective minor).
So, the cofactors are: a * det(| e f |), -b * det(| d f |), c * det(| d e |)
4. Add up the products:
determinant of A = a * det(| e f |) - b * det(| d f |) + c * det(| d e |)
This is the general process of finding the determinant using expansion by minors. It is a bit more involved than the method using diagonals, but it can be applied to matrices of any size.