# Thread: Determinant of a 3X3 matrix

1. ## Determinant of a 3X3 matrix

I am really confused and frustrated i put this matrix in a calculator and found the determinant to be 9 but when i do the math myself i get -11? What am i doing wrong here?? Here is the matrix:

1 -2 1
2 2 -1
1 -1 2

Thanks

2. bet you show your work and we'll spot what you're doing wrong.

3. $det\left(\begin{array}{ccc}1&-2&1\\2&2&-1\\1&-1&2\end{array}\right)$
$= (1*2*2) + (-2*-1*1) + (1*2*-1) - (1*2*1) - (-1*-1*1) - (2*2*-2) = 9$ by using the diagonals method.
to do this, write like this:
$det\left|\begin{array}{ccc}1&-2&1\\2&2&-1\\1&-1&2\end{array}\right|\begin{array}{cc}1&-2\\2&2\\1&-1\end{array}$

You multiply the first element by the one on the down sloping diagonal, and so forth.
element (1,1) * element (2,2) * element (3,3)
element (2,1) * element (3,2) * element (4,3)
and so on.

When you go up a diagonal, you subtract the product from the total.

Hope this helps!

4. That method, unfortunately, only works for 3 by 3 determinants.

Another way to do it- expand by minors on the first row:
$\left|\begin{array}{ccc}1 & - 2 & 1 \\ 2 & 2 & -1 \\ 1 & -1 & 2 \end{array}\right|= 1 \left|\begin{array}{cc}2 & -1 \\ -1 & 2\end{array}\right|-$ $(-2)\left|\begin{array}{cc}2 & -1 \\ 1 & 2\end{array}\right|$ $+ 1\left|\begin{array}{cc}2 & 2 \\ 1 & -1\end{array}\right|$

Yet another- row reduce:
$\begin{pmatrix}1 & -2 & 1 \\ 2 & 2 & -1\\ 1 & -1 & 2\end{pmatrix}$

Subtract twice the first row from the second row and subtract the first row from the third row to get
$\begin{pmatrix}1 & -2 & 1 \\ 0 & 6 & -3 \\ 0 & 1 & 1\end{pmatrix}$

Swap the second and third rows, then subtract 6 times that new second row from the new third row to get
$\begin{pmatrix}1 & -2 & 1 \\ 0 & 1 & 1 \\0 & 0 &-9\end{pmatrix}$

Adding (or subtracting) a multiple of one row from another does not change the determinant and swapping two rows multiplies the determinant by -1 so the determinant of the original matrix is -1 times the determinant of this "upper triangular matrix".

(Multiplying or dividing one row by a number, a row operation not used here, multiplies of divides the determinant by that number.)