What is the formula for determinant of a 3x3 matrix?

Answer 1

To answer this question, let me establish an example 3 x 3 matrix named "A":

A= [a b c]

[d e f]

[g h i]

The formula I will give you, called co-factor expansion, works for any size square matrix, so you could use it to find the determinant of a 2 x 2, 3 x 3, all the way up to an n x n matrix. To find the determinant, pick any row or column in the matrix. It will make your work much easier if you choose a row or column that has many zeroes in it.

A general notation that is often used to find the determinant of a matrix is to use straight bars in place of the brackets surrounding the matrix contents. So, if I was to say mathematically that I was finding the determinant of the above example matrix, I could write it as:

det(A)= |a b c|

|d e f|

|g h i|

This notation will be used in the formula, so it is important to know this.

For the sake of an arbitrary example, let us suppose I chose Row 1 of the matrix as my chosen row. To find the determinant of this matrix, I will perform the following calculation:

(-1)2(a)|e f| + (-1)3(b)|d f| + (-1)4(c)|d e|

|h i| |g i| |g h|

This is the specific application of this general formula to the example matrix:

(-1)i+j(aij)det(A1)

In this formula, i and j are the row and column addresses, respectively, of a given matrix element. So, like in our specific application, when Row 1 was chosen as our subject row, the first term was (-1)1+1(A11)det(A1). The element "a" is in the first row, first column of the matrix, mean i=1 and j=1, therefore the superscript of (-1) is 1+1=2. A11 is simply the value held in the address i=1, j=1 of the matrix A. For this application, A11 was "a". det(A1) is the determinant of the submatrix A1. This submatrix has no formal nomenclature, I simply call it this for ease of explanation. A1 is the matrix created by "crossing out" the row and column that belong to the matrix element A11. In this application, that means it is the submatrix that is left after crossing out a, b, c, d, and g, which is simply the 2 x 2 matrix e,f;h,i. Performing this same process for the remainder of the matrix elements in Row 1 will yield the determinant of the matrix. So, the "generalized" form of the specific application above is:

(-1)1+1(A11)det(A1) + (-1)1+2(A12)det(A2) + (-1)1+3(A13)det(A3)

where A1 is the submatrix created by crossing out Row 1 and Column 1, A2 is the submatrix created by crossing out Row 1 and Column 2, and A3 is the submatrix created by crossing out Row 1 and Column 3.

A final note is how to calculate the determinants of the submatrices. For a 3 x 3 matrix, its submatrices are all 2 x 2. For 2 x 2 matrices, a simple formula exists that makes this easy:

|a b| = (ad) - (bc)

|c d|

For higher-dimension matrices, the submatrices also become larger, making the computation much more intensive.