# Math Help - Gaus Jordan ELimination compared to cofactor inverse matrix

1. ## Gaus Jordan ELimination compared to cofactor inverse matrix

Hey guys i am stuck with the gaus jordan method of the following matrix.

A = 1 2 3
4 5 6
3 1 -2

For the co factor i worked out the C transpose as:

1/3 (-16 7 3
26 -8 6
-11 5 -3)

The gaus jordan is suppoesed to agree with my final cofactor matrix but i cant seem to understand how to get there?? i have donw the row eliminations to get the diagonal as 1 and the rest as 0.

Add (-4 * row1) to row2

Add (-3 * row1) to row3

Divide row2 by -3

Add (5 * row2) to row3

Divide row3 by -1

Add (-2 * row3) to row2

Add (-3 * row3) to row1

Add (-2 * row2) to row1

But now i don't understand what i have to do now?

2. Originally Posted by Latkan
Hey guys i am stuck with the gaus jordan method of the following matrix.

A = 1 2 3
4 5 6
3 1 -2

For the co factor i worked out the C transpose as:

1/3 (-16 7 3
26 -8 6
-11 5 -3)

The gaus jordan is suppoesed to agree with my final cofactor matrix but i cant seem to understand how to get there?? i have donw the row eliminations to get the diagonal as 1 and the rest as 0.

Add (-4 * row1) to row2

Add (-3 * row1) to row3

Divide row2 by -3

Add (5 * row2) to row3

Divide row3 by -1

Add (-2 * row3) to row2

Add (-3 * row3) to row1

Add (-2 * row2) to row1

But now i don't understand what i have to do now?
With the Gauss–Jordan method, for each row operation that you perform on A, you are supposed to carry out the same row operation starting with the identity matrix. This is best done by writing the identity matrix alongside A, and then performing the row operations on both. The process starts like this:

\begin{aligned}\begin{bmatrix}1&2&3&\vdots&1&0&0\\ 4&5&6&\vdots&0&1&0\\ 3&1&-2&\vdots&0&0&1\end{bmatrix} &\longrightarrow\begin{bmatrix}1&2&3&\vdots&1&0&0\ \ 0&-3&-6&\vdots&-4&1&0\\ 0&-5&-11&\vdots&-3&0&1\end{bmatrix}\\
&\longrightarrow\begin{bmatrix}1&2&3&\vdots&1&0&0\ \ 0&1&2&\vdots&\tfrac43&-\tfrac13&0\\ 0&-5&-11&\vdots&-3&0&1\end{bmatrix}.\end{aligned}

By the time that the left side of the double matrix has become the identity, the right side should give you the inverse matrix for A, like this:

$\begin{bmatrix}1&0&0&\vdots&-\tfrac{16}3&\tfrac73&-1\\ 0&1&0&\vdots&\tfrac{26}3&-\tfrac{11}3&2\\ 0&0&1&\vdots&-\tfrac{11}3&\tfrac53&-1\end{bmatrix}.$

Finally, you should always multiply the inverse matrix by the original matrix to check that the product is the identity matrix. If it isn't, that tells you that you have made a mistake somewhere.