Computing Determinants

Author:Ku, Yin Bon (Albert)

Determinant of 2 x 2 matrices

We already know that for any matrix

, we define

to be the signed area of the parallelogram formed by the column vectors of

. Now we will show that

. The proof is based on the following important properties of determinant: Let

be column vectors in

. Then

for any real number

(1) is obviously true because the parallelogram formed by the standard basis is a unit square. (2) is the convention we use to assign the sign to the area of the parallelogram formed by the column vectors of the matrix. (3) is obviously true when

because scaling the length of a side of a parallelogram will scale the area of the parallelogram by the same factor. When

, the area will be scaled by factor

and the sign of the area will be changed because the vectors

and

are pointing in the opposite direction. Therefore, the result in (3) still holds. Try the following applet to see why (4) holds. Remarks:

From (2), we can easily see that for any vector .
(3) and (4) together imply that any determinant is "linear" in its first column vector. Using (2), we can deduce that any determinant is also "linear" in its second column vector.
Other commonly-used notations for determinant: or .

Now, we are ready to prove the formula for the determinant of 2 x 2 matrices:

This completes the proof.

Determinant of 3 x 3 matrices

We can use the same idea to derive a formula for the determinant of any 3 x 3 matrix

. We define

to be the signed volume of the parallelepiped formed by the column vectors

and

. Then it has the following properties:

for any real number

Therefore, we can compute the determinant as follows:

This formula looks a bit complicated. Let's rewrite it into the one that is easier to memorize.

Computing Determinants

Determinant of 2 x 2 matrices

Determinant of 3 x 3 matrices

Exercise

New Resources

Discover Resources

Discover Topics