# Thread: Linear Algebra: Finding Determinant Using the Cramer's Ruler

1. ## Linear Algebra: Finding Determinant Using the Cramer's Ruler

1) Let A be a 4 by 4 matrix. If

2 0 0 0
0 2 1 0
0 4 3 2
0 -2 -1 2

find A.

Answer from back of the book:

A=
1 0 0 0
0 4 -1 1
0 -6 2 -2
0 1 0 1

*I have no clue on how to go from Adjoint A to A.

2. Originally Posted by googoogaga

1) Let A be a 4 by 4 matrix. If

2 0 0 0
0 2 1 0
0 4 3 2
0 -2 -1 2

find A.

Answer from back of the book:

A=
1 0 0 0
0 4 -1 1
0 -6 2 -2
0 1 0 1

*I have no clue on how to go from Adjoint A to A.
You may find this site helpful.

-Dan

3. I still don't understand because the matrix in the site is a 3 by 3 matrix. I just can't see it. Please I would like to see the steps. thanks

4. Originally Posted by googoogaga
2 0 0 0
0 2 1 0
0 4 3 2
0 -2 -1 2
I believe that the adjoint of A is equal to A. So find the adjoint of the adjoint.

You have to create a 4 x 4 matrix using the cofactor determinants. So
$A_{11} = \left | \begin{matrix} 2 & 1 & 0 \\ 4 & 3 & 2 \\ -2 & -1 & 2 \end{matrix} \right | = 4$

Then do the same for the other 15 cofactors determinants, then take the transpose of the resulting matrix. The website says that this is the adjoint, which I'll let you verify for yourself.

-Dan