How to Do rref and Inverse Matrix on the TI-89

If you are looking up how to do rref on the TI-89, you probably have an augmented matrix and a deadline. The TI-89 Titanium row reduces, inverts and analyses matrices with exact fractions, and this article covers the four commands most homework needs: rref(, the inverse, det( and the eigenvalue commands.

Type the matrix first

Every command below takes a matrix in square brackets. Separate the entries of a row with commas and the rows with semicolons: [1,2;3,4] is a 2 × 2 matrix. The opening bracket is 2nd with the comma key, the closing bracket is 2nd ÷, the semicolon is 2nd 9. The calculator echoes the matrix as [[1,2][3,4]], and you can type that form too.

2nd,1,22nd93,42nd÷→ [1,2;3,4]

If you need the same matrix several times, store it once with STO▶, for example [1,2;3,4]→a, and use a from then on. Use a letter that is not x or y, so it does not get in the way of solve( later.

STO▶→ store

How to do rref on the TI-89

All matrix commands sit in the Matrix submenu of MATH. Press 2nd 5 to open MATH, then 4 for Matrix, and pick rref( from the list. You can also type the name with alpha or find it in CATALOG.

2nd54then pick rref(→ rref(

Close the parenthesis after the matrix and press ENTER. rref( returns the reduced row echelon form; ref( stops at the row echelon form, which is what some courses ask for when they want to see the forward elimination.

Row reduction with rref( and ref(
Input	Result
rref([[1,2,3][4,5,6]])	10−1012
rref([1,2;3,4])	1001
ref([[2,4,6][1,1,1]])	123012

Read the first row as a system: x + 2y = 3 and 4x + 5y = 6 reduce to x = −1 and y = 2. A square matrix that reduces to the identity, as in the second row, is invertible. A row of zeros with a nonzero last entry means the system has no solution; a row of only zeros means a free variable.

Row operations one at a time

When the assignment asks for the steps, not just the final form, use the single row operations from the same menu: rowSwap( swaps two rows, mRow( multiplies one row, and mRowAdd( adds a multiple of one row to another.

Single row operations
Input	Result
rowSwap([[1,2,3][4,5,6][7,8,9]],1,3)	789456123
mRow(3,[[1,2][3,4]],2)	12912
mRowAdd(-2,[[1,2][3,4]],1,2)	1210

The last row is the classic elimination step: −2 times row 1 added to row 2.

Inverse matrix on the TI-89

The TI-89 Titanium writes the inverse as a power. Type the matrix or its variable, then ^, the (−) key and 1. Use (−), not the subtraction key, or you get a syntax error.

^(−)1→ ^-1

Calc89 accepts the same input and also the command inverse(. The result keeps fractions exact, so you can copy it into your work without rounding.

Inverse and identity
Input	Result
inverse([1,2;3,4])	−213/2−1/2
identity(3)	100010001

The first row is the same as [1,2;3,4]^-1. identity( builds the identity matrix of a given size, which is what a matrix times its inverse returns. If a matrix is singular, the calculator reports an error instead of an inverse.

Determinant with det(

det( is in the same Matrix submenu and takes a square matrix. A determinant of zero tells you before you try that the inverse does not exist. With a variable inside, det( gives a polynomial. Calc89 also has charPoly(, which builds the characteristic polynomial directly.

Determinant and characteristic polynomial
Input	Result
det([[1,2][3,4]])	−2
charPoly([1,2;3,4],x)	x²−5·x−2

Eigenvalues and eigenvectors

eigVl( returns the eigenvalues as a list, eigVc( the eigenvectors as the columns of a matrix, in the same order as the list and scaled to length 1. Your textbook may scale them differently; any nonzero multiple of an eigenvector is still an eigenvector.

Eigenvalues and eigenvectors
Input	Result
eigVl([2,0;0,3])	{2,3}
eigVc([2,0;0,3])	1001

A diagonal matrix is a good first test: the eigenvalues are the diagonal entries and the eigenvectors are the unit vectors.

Common mistakes

Commas instead of semicolons. [1,2,3,4] is one row, not a 2 × 2 matrix.
Rows of different length. They give a Dimension error; so does multiplying matrices whose inner sizes differ.
Subtraction in ^-1. The exponent needs the (−) key.
Decimal entries. Typing 0.5 instead of 1/2 makes the whole result approximate. Enter fractions to keep rref( exact.

In Calc89

Calc89 is a CAS calculator app intended for use with TI-89 Titanium. You type matrices with the same brackets and semicolons, and rref(, det(, ^-1, eigVl( and eigVc( give the exact results shown above. Typing a 4 × 4 matrix on a small screen is tedious, so point the camera scanner at the matrix in your book instead. The keystroke tutor then shows you the menu path and every key for the TI-89 Titanium, so you can repeat the steps on the handheld in class. Calc89 is free to download, Pro features as an in-app purchase.

To solve a system with these tools, see Systems of equations on the TI-89 Titanium; for transpose, augment( and more, the guide Linear algebra on the TI-89 Titanium.

Frequently asked questions

How do I do rref on the TI-89 Titanium?

Open the Matrix submenu with 2nd 5 and 4, pick rref(, type the matrix in brackets and close the parenthesis. For example rref([[1,2,3][4,5,6]]) returns [[1,0,−1][0,1,2]].

How do I find an inverse matrix on the TI-89?

Type the matrix or its variable, then ^, the (−) key and 1, and press ENTER. A singular matrix gives an error instead of a result.

Can the TI-89 find eigenvalues?

Yes. eigVl( returns the eigenvalues as a list and eigVc( the unit length eigenvectors as the columns of a matrix.

Matrices on the TI-89: rref, Inverse, Determinant and Eigenvalues