# TI-BASIC:Det

Command Summary

Calculates the determinant of a square matrix.

Command Syntax

det(matrix)

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(.

TI-83/84/+/SE

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.

## Contents

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

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

For larger matrices, the determinant can be computed using the [wikipedia:Laplace_expansion 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 ${\displaystyle O\left(n!\right)}$ operations, the method usually used in calculators is Gaussian elimination, which only needs ${\displaystyle O\left(n^{3}\right)}$ operations.

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

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

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