# Knowledge Base

## Solution 11569: Algorithm Used for Finding the Determinant of a Square Matrix on the TI-81, TI-82, TI-83, TI-85, or TI-86.

### How is the determinant of a square matrix calculated on the TI-81, TI-82, TI-83, TI-85, and TI-86?

Many methods exist for computing the determinant of a square matrix. The one used on the TI-81, TI-82, TI-83, TI-85, and TI-86 is a byproduct of the LU-Factorization method for solving a set of linear equation of the form:

Ax = b

where, A is a square matrix, b is a column vector and x is a column vector of unknowns.

The LU-Factorization method relies on the fact that a square matrix can be written as the product of a lower triangular matrix, L, containing ones along the main diagonal and an upper triangular matrix, U. That is,

A = L*U

To illustrate this point, consider the matrix,

A = [[ 2,-1,0][4,3,-3][8,-14,5]]

which can also be expressed as:

A = [[1,0,0][2,1,0][4,-2,1]]*[[2,-1,0][0,5,-3][0,0,-1]]

Hence, L and U for this problem are:

L = [[1,0,0][2,1,0][4,-2,1]]

U = [[2,-1,0][0,5,-3][0,0,-1]]

Once in this form, one may compute the det(A) as follows using the properties of determinants.

det(A) = det(L)*det(U)

The determinant of a triangular matrix can be computed by multiplying the elements along its main diagonal. Since L always has ones along its main diagonal, det(L) = 1. Now, simplify the previous equation as follows and compute:

det(A) = det(U) = (2)(5)(-1) = -10

Although only a 3 by 3 matrix was used in this example, this method is applicable in general to any square matrix. It also has excellent accuracy characteristics while being one of the more computationally efficient algorithms, which is why it was selected.