Inverse of 3X3 Matrix

[Jason76]

This one is very important. I hope I can get some hints. As far as finding the inverse matrix of a 3X3 Matrix, I got everything else down pat, regarding the cofactor method:

 

[HallsofIvy]

You have the words "Take Determinant" but say nothing about the determinant. You found that the determinant is 10, right?

The calculations on the left are 4 of the nine minors: In the first, you have crossed out the top row and left column and got "2". As a result, the number in the top row and left column of the matrix on the right is "2". In the second, you have crossed out the top row (a little bent!) and middle column and got "2". As a result, the number in the top row and middle column of the matrix on the right is "2". In the third, you have jumped down and crossed out the bottom row and middle column and got "-10". As a result, the number in the bottom row and middle column of the matrix on the right is "-10". Finally, you have crossed out the bottom row and right column and got "0". Now, the number in the bottom row and right column of the matrix on the right is "1"! That's wrong! The matrix of minors should be
\(\displaystyle \begin{bmatrix}2 & 2 & 2 \\ -2 & 3 & 3 \\ 0 & -10 & 0 \end{bmatrix}\).

If you were to divide each of those by 10, the determinant, you would get the inverse to the original matrix.

 

[Jason76]

Sorry for mistake in copying the problem from a website. Here is the correct copy:

 

[Jason76]

Ok, so now we got 2 on the top row left column, 2 on the top row middle column, -10, bottom row, middle column, 0 on the bottom row right column. But how did we get the other numbers on the new matrix? What numbers are crossed out to make the other minors? For some reason, the website didn't list out all "crossing out" patterns.

 

[Deleted member 4993]

Ok, so now we got 2 on the top row left column, 2 on the top row middle column, -10, bottom row, middle column, 0 on the bottom row right column. But how did we get the other numbers on the new matrix? What numbers are crossed out to make the other minors? For some reason, the website didn't list out all "crossing out" patterns.

Yes, they did not - hoping that "you" will find those!

For example,

did you try to find the co-factor of A(2,2)? You will need to cross-out the numbers in the 2nd row and the second column.

What is it? What is the determinant of that co-factor?

 

[Jason76]

I have one last question. It's on the chart: Note: Little mistake. The (0 - 0) next to the first chart, should be (0 * 0)


I pretty much got the idea of this. Wherever the two lines meet, that place becomes where the determinant is placed on the new "matrix of determinant answers".

 

[HallsofIvy]

I think that the one you have not yet done is the minor corresponding to the middle number: cross out the middle row and middle column leaving
\(\displaystyle \left|\begin{array}{cc}3 & 2 \\ 0 & 1 \end{array}\right|= 3\)

(0ne point I misspoke before- the inverse is NOT the array of minors divided by the determinant, it is the adjoint of the array of minors. That is, rows become columns, columns become rows. )

 

[Jason76]

I think that the one you have not yet done is the minor corresponding to the middle number: cross out the middle row and middle column leaving
\(\displaystyle \left|\begin{array}{cc}3 & 2 \\ 0 & 1 \end{array}\right|= 3\)

(0ne point I misspoke before- the inverse is NOT the array of minors divided by the determinant, it is the adjoint of the array of minors. That is, rows become columns, columns become rows. )

Ok, cool..