Determinant of a matrix using cofactor expansion - confused

• Dec 13th 2010, 11:09 PM
Kakariki
Determinant of a matrix using cofactor expansion - confused
Hey! I am studying for my linear algebra exam and am having some trouble with determining the determinant of a square matrix using cofactor expansion.

Here is an example of what I am having trouble with:
Question
$\displaystyle \displaystyle \begin{bmatrix} 0 & 3 & 1\\ 1 & 1 & 2\\ 3 & 2 & 4\\ \end{bmatrix}$

Attempted solution
I know that the determinant of this matrix is 5, I just can't seem to get that answer in what I am doing. I will use the first row in this attempt:
$\displaystyle 0 * \displaystype \begin{bmatrix} 1 & 2\\ 2 & 4\\ \end{bmatrix}$
$\displaystyle + 3 * \displaystype \begin{bmatrix} 1 & 2\\ 3 & 4\\ \end{bmatrix}$
$\displaystyle + 1 * \displaystype \begin{bmatrix} 1 & 1\\ 3 & 2\\ \end{bmatrix}$
then replacing those 2x2 matrices with their determinants gives:
$\displaystyle (0 * 0) + (3 * -2) + (1 * -1)$
$\displaystyle = -7$
Which is NOT 5. I tried this with other columns and rows and am getting different answers for those. Am I doing this wrong or does it not work on some matrices???

Hopefully someone can help! Thanks for looking :)
• Dec 13th 2010, 11:29 PM
earboth
Quote:

Originally Posted by Kakariki
Hey! I am studying for my linear algebra exam and am having some trouble with determining the determinant of a square matrix using cofactor expansion.

Here is an example of what I am having trouble with:
Question
$\displaystyle \displaystyle \begin{bmatrix} 0 & 3 & 1\\ 1 & 1 & 2\\ 3 & 2 & 4\\ \end{bmatrix}$

Attempted solution
I know that the determinant of this matrix is 5, I just can't seem to get that answer in what I am doing. I will use the first row in this attempt:
$\displaystyle 0 * \displaystype \begin{bmatrix} 1 & 2\\ 2 & 4\\ \end{bmatrix}$
- $\displaystyle 3 * \displaystype \begin{bmatrix} 1 & 2\\ 3 & 4\\ \end{bmatrix}$
$\displaystyle + 1 * \displaystype \begin{bmatrix} 1 & 1\\ 3 & 2\\ \end{bmatrix}$
then replacing those 2x2 matrices with their determinants gives:
$\displaystyle (0 * 0) \bold{-} (3 * -2) + (1 * -1)$
$\displaystyle = 5$
Which is NOT 5. I tried this with other columns and rows and am getting different answers for those. Am I doing this wrong or does it not work on some matrices???

Hopefully someone can help! Thanks for looking :)

The sign of the co-factors is changing from column to column (or row to row).

See my corrections.
• Dec 13th 2010, 11:45 PM
Kakariki
I looked back over my notes and I seemed to have somehow missed the part explaining the pattern of sign changing... With that I understand how to go about doing this!!! Thank you so much!