311.3.1.1 - Determinants And Cofactor Expansion

In summary, the conversation discusses using cofactor expansion to compare determinants and compute the determinant of a matrix. It also includes an example of using this method to calculate the determinant of a matrix. The method is explained and the importance of giving threads descriptive titles is mentioned.
  • #1
karush
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311 Determinants And Cofactor Expansion (3.1.1)

a. Compare the determinants using a cofactor expansion across the first row.

b. compute the determinant by a cofactor expansion down the second column.

$$\left|
\begin{array}{rrr}
3&0& 4\\
2&3& 2\\
0&5&-1\\
\end{array}
\right|$$

ok I accually start this class tomorro but thot I would try some basic stuff
already stuck...
 
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  • #2
Re: 311.3.1.1

karush said:
a. Compare the determinants using a cofactor expansion across the first row.

b. compute the determinant by a cofactor expansion down the second column.

$$\left|
\begin{array}{rrr}
3&0& 4\\
2&3& 2\\
0&5&-1\\
\end{array}
\right|$$

ok I accually start this class tomorro but thot I would try some basic stuff
already stuck...

(a)

$3\begin{vmatrix}
3 & 2\\
5 & -1
\end{vmatrix} - 0\begin{vmatrix}
2 &2 \\
0 & -1
\end{vmatrix}+4\begin{vmatrix}
2 & 3\\
0& 5
\end{vmatrix} =3(-3-10)-0(-2-0)+4(10-0)=1$

(b)

note $\begin{vmatrix}
3 & 0 & 4\\
2 & 3 & 2\\
0 & 5& -1
\end{vmatrix}=\begin{vmatrix}
0 & 4 & 3\\
3 & 2 & 2\\
5 & -1 & 0
\end{vmatrix}$

the second determinant is formed by moving the 1st column in the original matrix to the 3rd column. Now use the new 1st column and its cofactors ...

$0\begin{vmatrix}
2 & 2\\
-1 & 0
\end{vmatrix}-3\begin{vmatrix}
4 & 3\\
-1 & 0
\end{vmatrix}+5\begin{vmatrix}
4 &3 \\
2 & 2
\end{vmatrix}=0[0-(-2)]-3[0-(-3)]+5(8-6)=1$

This is a method I learned in undergrad physics that "stuck". There are other methods to do these ... I'm sure someone else well-versed in linear algebra will contribute.
 
  • #3
Please give threads titles that briefly describe the thread topic. A thread title that is simply a series of digits and periods is not what we consider a good thread title. :)
 
  • #4
https://dl.orangedox.com/GXEVNm73NxaGC9F7Cy

SSCwt.png
 

FAQ: 311.3.1.1 - Determinants And Cofactor Expansion

1. What is the purpose of using determinants and cofactor expansion?

Determinants and cofactor expansion are used to find the solutions of systems of linear equations, calculate the area/volume of a figure, and determine the invertibility of a matrix.

2. How is the determinant of a matrix calculated?

The determinant of a matrix is calculated by multiplying the elements in the main diagonal and subtracting the product of the elements in the other diagonal, following a specific pattern depending on the size of the matrix.

3. What is the role of cofactors in determining the determinant?

Cofactors are used to simplify the calculation of the determinant by reducing the size of the matrix and applying the same process recursively. They are also used to find the inverse of a matrix.

4. Can determinants and cofactor expansion be used for any size of matrix?

Yes, determinants and cofactor expansion can be used for any square matrix, regardless of its size. However, the calculation process becomes more complex as the size of the matrix increases.

5. What are the applications of determinants and cofactor expansion in real life?

Determinants and cofactor expansion have various applications in fields such as physics, engineering, economics, and computer science. They are used to analyze systems of equations, solve optimization problems, and model real-life scenarios.

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