math111_logo Geometry of Determinant

2. Determinant of 3 by 3 matrix

The determinant of a 3 by 3 matrix is

det[ a11 a12 a13 ] = a11a22a33 + a12a23a31 + a13a21a32 - a13a22a31 - a11a23a32 - a12a21a33.
a21 a22 a23
a31 a32 a33

A simple way of memorizing the formula is to repeat the first and the second columns to the right of the matrix and then multiply as follows to get the six terms.

Similar to the 2 by 2 case, the absolute value of the determinant of a 3 by 3 matrix is the volume of a parallelogram spanned by the three column vectors. The claim may be verified in a way similar to the 2 by 2 case. However, this involves complicated 3 dimensional picture and we will not carry this out here. See this exercise for a very special case. We will provide a more general explaination later on. For the moment, we will simply accept the interpretation.

A consequence of the interpretation is that

det[u v w] = 0
u v w lie on a plane (because the volume is zero)
u v w are linearly dependent (see this discussion)
⇔ [u v w] is not invertible. (by this result)

This clearly generalizes the criterion for the invertibility of a 2 by 2 matrix. The criterion will still be true for general n by n matrices, as long as the absolute value of the general determinant still measures the volume of the parallelogram.

Example The volume of the tetrahedron with the vertices (1, -2, 1), (2, 1, 2), (0, -1, 4), (2, -3, 1) is one sixth of the volume of the parallelogram spanned by u = (2, 1, 2) - (1, -2, 1) = (1, 3, 1), v = (0, -1, 4) - (1, -2, 1) = (-1, 1, 3), w = (2, -3, 1) - (1, -2, 1) = (1, -1, 0), which is

(1/6)| det[ 1 -1 1 ] | = (1/6)|1×1×0 + (-1)×(-1)×1 + 1×3×3 - 1×1×1 - 1×(-1)×3 - (-1)×3×0| = 2.
3 1 -1
1 3 0

Similar to the 2 by 2 case, the sign of the determinant of a 3 by 3 matrix is also determined by the orientation of the three column vectors. Specifically,

u, v, w follows the right hand rule ⇒ det[u v w] ≥ 0
u, v, w follows the left hand rule ⇒ det[u v w] ≤ 0

The reason for the orientation to determines the sign is similar to the 2 by 2 case. We may construct a (somewhat more complicated) movie to deform the given arrangement into one of the two standard arrangements.

The extension of the determinant to matrices of bigger size should reflect the requirements that the absolute value is the volume of a parallelogram and the sign should be the orientation of the arrangement of the column vectors. We emphasis that the following naive extension to 4 by 4 matrices does not satisfy the requirements.

In fact, the determinant of a 4 by 4 matrix contains 24 terms. In general, the determinant of an n by n matrix contains n! = 1×2×3×...×n terms.


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