The det( Command

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Command Summary

Calculates the determinant of a square matrix.

Command Syntax

det(matrix)

Menu Location

Press:

1. MATRX (83) or 2nd MATRX (83+ or higher) to access the matrix menu
2. LEFT to access the MATH submenu
3. ENTER to select det(.

Calculator Compatibility

TI-83/84/+/SE

Token Size

1 byte

The det( command calculates the determinant of a square matrix. If its argument is not a square matrix, ERR:INVALID DIM will be thrown.

Advanced Uses

If [A] is an N×N matrix, then the roots of det([A]-X identity(N)) are the eigenvalues of [A].

Formulas

For 2×2 matrices, the determinant is simply

(1)

\begin{align} \det\left( \begin{bmatrix} a & b\\c & d \end{bmatrix} \right) = \begin{vmatrix} a & b\\c & d \end{vmatrix} = ad-bc \end{align}

For larger matrices, the determinant can be computed using the Laplace expansion, which allows you to express the determinant of an n×n matrix in terms of the determinants of (n-1)×(n-1) matrices. However, since the Laplace expansion takes $O\left( n! \right)$ operations, the method usually used in calculators is Gaussian elimination, which only needs $O\left( n^3 \right)$ operations.

The matrix is first decomposed into a unit lower-triangular matrix and an upper-triangular matrix using elementary row operations:

(2)

\begin{pmatrix} {1}&{}&{}\\ {\vdots}&{\ddots}&{}\\ {\times}&{\cdots}&{1}\end{pmatrix} \begin{pmatrix}{\times}&{\cdots}&{\times}\\ {}&{\ddots}&{\vdots}\\ {}&{}&{\times} \end{pmatrix}

The determinant is then calculated as the product of the diagonal elements of the upper-triangular matrix.

Error Conditions

• ERR:INVALID DIM is thrown when the matrix is not square.

Related Commands

• identity(
• ref(
• rref(

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