| Property | Value |
|---|---|
| Hex Value | $B3 |
| Categories | |
| Localizations |
|
det(
Overview
Returns determinant of matrix.
Availability: Token available everywhere.
Syntax
det(matrix)
Arguments
| Name | Type | Optional |
|---|---|---|
| matrix | matrix |
Location
2nd, matrix, MATH, 1:det(
Description
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
Source: parts of this page were written by the following TI|BD contributors: burr, DarkerLine, GoVegan, Myles_Zadok, thornahawk.
History
| Calculator | OS Version | Description |
|---|---|---|
| TI-82 | 1.0 | det added |
| TI-83 | 0.01013 | Renamed det to det( |