Linear algebra on the TI-89 Titanium: matrices, det, rref and eigenvalues

Entering a matrix

A matrix is typed inside square brackets. Elements in a row are separated by commas, rows by semicolons. The brackets are 2nd with the comma key and 2nd ÷, the semicolon is 2nd 9. So the 2 × 2 matrix with rows 1, 2 and 3, 4 is [1,2;3,4]. The calculator shows it back as [[1,2][3,4]], and you can type that form too.

To reuse a matrix, store it in a variable with STO▶: [1,2;3,4]→a. After that, det(a) or a^-1 work like any other expression. For large matrices the Data/Matrix Editor in APPS is easier than one long line.

The Matrix menu

All matrix commands live in MATH, which is 2nd 5. Item 4 opens the Matrix submenu, where you pick the command. Every command below can also be typed with the alpha letters or picked from CATALOG.

Determinant with det(

det( takes a square matrix. With symbols inside, the result is a polynomial, which is how you set up a characteristic equation by hand.

Row reduction with rref( and ref(

rref( returns the reduced row echelon form, ref( the row echelon form. Both keep fractions exact. For a system of equations, row reduce the augmented matrix and read the solution from the last column.

The second row is the augmented matrix of x + 2y = 3 and 4x + 5y = 6. The reduced form reads x = −1 and y = 2. When a row of zeros appears with a nonzero last entry, the system has no solution; a row of only zeros means a free variable.

Inverse and transpose

On the TI-89 Titanium the inverse is a power: type the matrix, then ^ (−) 1. The transpose ᵀ is in the same Matrix menu and goes after the matrix. Calc89 accepts both forms and also the command names inverse( and transpose(.

The first row is the same as [1,2;3,4]^-1. A singular matrix has no inverse, and the calculator reports an error instead of a result; check det( first if you are not sure.

Eigenvalues and eigenvectors

eigVl( returns the eigenvalues as a list, eigVc( the eigenvectors as the columns of a matrix, scaled to unit length. The order of the columns matches the order of the list.

For a diagonal matrix the eigenvalues are the diagonal entries and the eigenvectors are the unit vectors, which makes it a good first test that you typed the command correctly.

Systems of linear equations

There are three ways to the same answer. solve( takes the equations joined with and and a list of unknowns in braces (2nd ( and 2nd )): solve(x+y=3 and x-y=1,{x,y}) finds the same solution, x = 2 and y = 1, as the first two rows below. simult( takes the coefficient matrix and the column of constants. Calc89 also accepts linSolve( with a list of equations. augment( builds the augmented matrix for rref(.

Common mistakes

• Mismatched sizes. Rows with different lengths, or a product of matrices whose inner sizes differ, give a Dimension error.
• Comma instead of semicolon. [1,2,3,4] is one row with four entries, not a 2 × 2 matrix.
• Minus instead of (−) in ^-1. Use the (−) key for the negative exponent.
• Old variables. A stored x or y turns solve( into a check of numbers. Run NewProb (2nd F1) first.

FAQ

Calc89 is an independent app intended for use with TI-89 Titanium. It is not affiliated with or endorsed by Texas Instruments. TI-89 Titanium is a trademark of Texas Instruments.