# Matrix Operations on a TI-83 Graphing Calculator

# Matrix Operations on a TI -83 Graphing Calculator

(This paper is based on a talk given in Spring Semester
2004.) The use of a graphing calculator can

be useful and convenient, especially when reducing a matrix that has entries
with many decimal

places. The inverse of a matrix can also be found easily. One of the homework
assignments for

MAT 119 is to reduce a matrix with a graphing calculator.

## Introduction

Two important applications with matrices in MAT 119 are
solving a system of linear equations and finding

the inverse of a matrix. The TI series of graphing calculators is able to find
the inverse and put a matrix in

reduced row echelon form automatically, but students still should learn how to
do row operations, especially

when using the Simplex Method in Chapter 4. The purpose of this paper is to
indicate the appropriate steps.

A brief word on notation. When a box is drawn around a
symbol, that means to press that key on

the TI. Hence means to push the
multiplcation button . If two key presses are used, they will be to

refer to functions printed in different colored keys on the calcuator. The
keypresses will be given, along

with the operation to be performed, the latter in slanted type and in brackets.
For instance,

means to press then
and this accesses the function ex; this is
in yellow ink on the TI-83. Also,

[A] puts the calculator in alphabetic mode
and enters the letter A in the display; these letters

are printed in green ink on the TI-83.

## Solving A System of Linear Equations

Suppose you want to solve the system.

25x + 61y - 12z = 10

18x - 12y + 7z = -9

3x + 4y - z = 12

Here's how to solve it.

1. Enter the augmented matrix into the calculator.

Press to put the
calculator in MATRIX mode.

You now see a list of matrix names down the left-hand side of the screen, and
the words NAMES,

MATH and EDIT across the top, with NAMES currently "highlighted." Pressing the
right-arrow key

or will change which word is highlighted.
Pressing will select the name of the matrix
you

want to enter. Let's enter the system above into the matrix A; move the cursor
to the right so that EDIT

is highlighted (press if you haven't moved
the highlighted part) and press

You are now prompted to enter the dimensions of the matrix A; enter the number
of rows

and the number of columns The TI-83 can
handle matrices with up to 99 rows and/or columns,

but cannot handle a matrix with both due to limited memory.

NOTE. At any time, if you realize you made a mistake, you
can press [QUIT] and then start over.

Now you need to enter the coefficients and numbers on the right hand side. The
cursor is placed in the first

row, first column of the matrix. If you type in a number (or an expression) and
hit it will put that

number into the matrix and move to the next position, to the right (if you're
not at the right-most column)

or down to the first column of the next row. After entering a number in the last
row and last column, the

cursor stays there.

So we need to put in the coefficients. The keystrokes for
this are

etc.

2. Perform the row operations.

Now you need to get back to the calculation mode. Press
[QUIT]. You are ready to perform

row operations on your matrix A.

The first thing we will do is to swap the first and third
rows of A. Press then
twelve

times; the cursor should be next to a command called rowSwap(. Press
, which will bring you back

to the "calculation mode." Now you need to enter the matrix. press
. (If you want to do a

row reduction on a matrix other than A, you need to "scroll down" by pressing
repeatedly.) Then press

This gives you the matrix
If you need to see columns further to the
right of

the screen, press the key.

NOTE. Doing row operations on A will not change the value
of A on a TI -83, so we will need to keep

track of the new matrix. We will do this by using the ANS
command. It is also a good idea

to copy onto paper the matrix we get at this point, in case we have to go back
to it.

Now we want to divide the first row by 3, so that there is
a 1 in the upper left corner. Press

which will display *row( in the calculator.
We want to multiply the first row of ANS by

1/3, so we type in. then
The matrix we get is

(Don't worry about the decimal approximation; we'll x it
at the very end.)

Now you need to subtract 18 times Row 1 from Row 2, to turn the entry in the 2nd
row, 1st column

into a 0. Enter.

The new matrix is

Now, you perform these same types of row operations over and over. The key sequences are below.

25 times row 1 from row 3).

row 2 by -36).

3.

(subtract 27.666666667 times row 2 from row
3).

4. If you press
repeatedly, you will find that the entry in the 3rd row and 3rd column is
approximately

6.32407407407. This is what you want to divide row 3 by.

5. The matrix is now in row echelon form. To get it into
reduced row echelon form, continue with.

Now the matrix is close to reduced row echelon form; the
entries on the main diagonal are very close to

1, and the other entries in the first three columns are close to zero. Press or
hold the until you can see

the fourth column of the matrix. The value in the first row is x = 4.566617845,
the value in the second row

is y = -6.443631037, and the value in the third row is z = -24.07467057.

You can try to convert these decimals to fractional form
by typing in [ANS]

[!Frac] In this case, it doesn't help - the
TI-83 only tries fractions with denominators less than

or equal to 100 - but it may in others.

## Inverting a Matrix

Inverting a matrix can be done on the TI without as much
work; it is built-in to the calculator. Here we

will invert the matrix

1. Enter the matrix into the calculator.

This can be done as in the previous section; put it into matrix B, then press
[QUIT] to get

back into "calculation mode."

2. Calculate the inverse.

Type in. You get,
after a few seconds have gone by.

Other matrix computations are possible. For instance, to find
, type

the answer is

## Matrix Operations on a TI-85 Graphing Calculator

(This paper is based on a talk given in Spring Semester
2004.) The use of a graphing calculator can

be useful and convenient, especially when reducing a matrix that has entries
with many decimal

places. The inverse of a matrix can also be found easily. One of the homework
assignments for

MAT 119 is to reduce a matrix with a graphing calculator.

## Introduction

Two important applications with matrices in MAT 119 are
solving a system of linear equations and finding

the inverse of a matrix. The TI series of graphing calculators is able to find
the inverse and put a matrix in

reduced row echelon form automatically, but students still should learn how to
do row operations, especially

when using the Simplex Method in Chapter 4. The purpose of this paper is to
indicate the appropriate steps.

A brief word on notation. When a box is drawn around a
symbol, that means to press that key on

the TI. Hence means to push the
multiplcation button. If two key presses are used, they will be to

refer to functions printed in different colored keys on the calcuator. The
keypresses will be given, along

with the operation to be performed, the latter in slanted type and in brackets.
For instance,

means to press , then
and this accesses the function ex; this is
in yellow ink on the TI-85. Also,

[A] puts the calculator in alphabetic mode
and enters the letter A in the display; these letters

are printed in blue ink on the TI-85.

## Solving A System of Linear Equations

Suppose you want to solve the system.

Here's how to solve it.

1. Enter the augmented matrix into the calculator.

Press [MATRX] to put
the calculator in MATRIX mode.

You will see five "words" above the keys
through NAMES, EDIT, MATH, OPS, CPLX. To

enter a matrix, press the key under EDIT You
are now asked for the name of the matrix, which can

be any single letter. The cursor is in alphabetic mode, so all you have to do is
press

You are now prompted to enter the dimensions of the matrix A; enter the number
of rows

and the number of columns The TI-85 can
handle matrices with up to 99 rows and/or columns,

but cannot handle a matrix with both due to limited memory.

NOTE. At any time, if you realize you made a mistake, you
can press and then start over.

Now you need to enter the coefficients and numbers on the right hand side. The
cursor is placed in the first

row, first column of the matrix. If you type in a number (or an expression) and
hit it will put that

number into the matrix and move to the next position, to the right (if you're
not at the right-most column)

or down to the first column of the next row. After entering a number in the last
row and last column, the

cursor stays there.

So we need to put in the coefficients. The keystrokes for
this are

etc.

2. Perform the row operations.

Now you need to get back to the calculation mode. Press
. You are

ready to perform row operations on your matrix A.

The first thing we will do is to swap the first and third
rows of A. Press the key to get the

following "words" above the keys through
aug, rSwap, rAdd, multR, and mRAdd. To swap
two

rows, press [rSwap(]. Now you need to type
the rest in. the name of the matrix (press ALPHA LOG

and the two rows to be swapped (press

This gives you the matrix
If you need to see columns further to the right of

the screen, press the key.

NOTE. Doing row operations on A will not change the value
of A on a TI-85, so we will need to keep

track of the new matrix. We will do this by using the ANS
command. It is also a good idea

to copy onto paper the matrix we get at this point, in case we have to go back
to it.

Now we want to divide the first row by 3, so that there is a 1 in the upper left
corner. We are still have the

options aug, rSwap, rAdd, multR, and mRAdd, so we can just press
[multR(]. We want to multiply the first

row of ANS by 1/3, so we type in. The matrix
we get is

(Don't worry about the decimal approximation; we'll x it
at the very end.)

Now you need to subtract 18 times Row 1 from Row 2, to turn the entry in the 2nd
row, 1st column

into a 0. Enter. F5 The new

matrix is

Now, you perform these same types of row operations over and over. The key sequences are below.

1. (subtract 25 times
row 1 from

row 3).

2. [multR(]
(divide row 2 by -36).

3. [mRAdd(]

(subtract 27.6666666667 times row 2 from row
3).

4. If you press repeatedly, you will find
that the entry in the 3rd row and 3rd column is approximately

6.32407407409. This is what you want to divide row 3 by.
[multR(]

5. The matrix is now in row echelon form. To get it into
reduced row echelon form, continue with.

[mRAdd(]

Now the matrix is close to reduced row echelon form; the
entries on the main diagonal are very close to

1, and the other entries in the first three columns are close to zero. Press or
hold the until you can see

the fourth column of the matrix. The value in the first row is x =
4.56661786066, the value in the second

row is y = -6.44363103925, and the value in the third row is z = -24.074670571.

You can try to convert these decimals to fractional form
by typing in

[MATH] In this case, this only gives you an
exact value for z,

namely -16443=683. The TI-85 only tries fractions with denominators less than or
equal to 1000, so you

may get exact values in other situations. Note that roundo errors kept us from
getting exact answers for x

(3119/683) and y (-4401=683).

## Inverting a Matrix

Inverting a matrix can be done on the TI without as much
work; it is built-in to the calculator. Here we

will invert the matrix

1. Enter the matrix into the calculator.

This can be done as in the previous section; put it into matrix B, then press
to get back into

"calculation mode."

2. Calculate the inverse.

Type in. You get
after a few seconds have

gone by.

Other matrix computations are possible. For instance, to
find , type

the answer is

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