TI Project: Linear Transformations 1
TI Project: Linear Transformations
The problems will help you to visualize the action of linear transformations on the plane, and to view
matrices as functions acting on vectors. Review Sections 1.8 and 1.9 of the text. Data for this project is in
the LINTRNS section of the PROJDATA program.
TI-83+ and TI-84+ (MP)
Operations.
[A]*[B] Matrix multiplication (2 > Y*2 > Ze), A and B are compatible for
multiplication.
TI-89
Operations.
aa*bb Matrix multiplication (aa p bb ¸). A and B are compatible for multiplication.
Graphing. Before you work on this assignment, you need to clear any graphs or drawings that were
displayed on your calculator screen. Press ¥ % 2 ˆ to clear the old drawings. Select “ ¥ #
and press † on every defined function and plot to deselect the graphs. Set your viewing window as in
Figure 1.
Figure 1: Window
TI Project: Linear Transformations 2
Problem 1. For each one of the following matrices consider the transformation Tx = Ax (there are four
transformations to consider). A =
10
02
⎡
⎣
⎢⎤
⎦
⎥
,
15
02
.
⎡
⎣
⎢⎤
⎦
⎥
,
−
⎡
⎣
⎢⎤
⎦
⎥
10
01
,
−−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
2
2
2
2
2
2
2
2
. The transformation T will change
your figure in a very specific way. You are to describe the effect of each linear transformation, the following steps
will guide you.
2. Use your calculator to perform the product A*E and store it in B. Write B down.
3. Explain why the operation executed in (d) corresponds to applying the transformation T to each one of the points
that determine the original figure.
5. Repeat (a), (b), (d), and (f) for the remaining matrices.
You may want to use the steps described at the beginning to clear some of the drawings, if they do not let you view
the new image clearly. If you followed the steps correctly, you should have the original data in E.
produced by multiplying F by a 2×2 matrix A and drawing the figure with the program DWMT.
(a) Find 2×2 matrices A1, A2, A3, and A4 that will produce the given pictures—starting from the
original. (Use shears, reflections, contractions, expansions and rotations—your final matrix may be a
TI Project: Matrix Algebra
In this project you will practice entering matrices directly into your calculator and using the matrix
editor. You will become acquainted with the matrix operations that are built into the TI Calculator,
although some of the operations are to be done by hand. You will also create random matrices and
operate on them to discover properties of matrices. Data for this project is in the MATRALG section of the
PROJDATA program.
Programs. You will need the RDIAG program to generate some matrices.
RDIAG Generates a diagonal matrix of a specified size with diagonal integer entries in the
interval [–9, 9]. Prompts for a positive integer—the size of the matrix—as the input.
TI-83+ and TI-84+ (MP)
Operations.
[A]+[B] Matrix addition (2 > Y+2 > Ze). A and B are compatible for addition.
r*[A] Scalar multiplication (r * 2 > Ye), r is a real number and A is a matrix.
[A]^n Matrix powers (2 > Y^ n e). A is a square matrix and n is a positive integer.
[A]T Computes the transpose of A (2 > Y2 > (right arrow to MATH menu) Ze).
randM(r,c) Returns as random matrix of size r × c with integer entries in the interval [–9, 9].
2 > (right arrow to MATH menu) 6e). Requires two arguments—row size and
TI-89
Operations.
aa+bb Matrix addition (aa + bb e). A and B are compatible for addition.
aa-bb Matrix subtraction (aa – bb e). A and B are compatible for subtraction.
TI Project: LU Factorization 2
Problem 1. Use the random matrix command to generate matrices of the sizes indicated below. Write these matrices
down and compute the indicated operations.
(a) A and B both of size 5 × 6, and C of size 6 × 5. Compute the following. If your calculator gives you an error
message discuss with your partner why the error occurred.
(b) A and C both of size 7 × 3, and B of size 3 × 4. Compare the following. If your calculator gives you an error
message discuss with your partner why the error occurred.
i. 5A – B
ii. B + (–7)B, (–6)B. Compare these results.
iii. A + 2C
iv. 1.5A – C
Problem 2. Read the Definition of matrix multiplication from this section. This definition tells you how to multiply
two matrices. Notice the restrictions on the sizes of the matrices involved. Let A =
26
04
12
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
and B =
1301
4523
⎡
⎣
⎢⎤
⎦
⎥
.
To compute AB, let b1, b2, b3, and b4 denote, respectively, the columns of the matrix B. You need to compute the
products Ab1, Ab2, Ab3 and Ab4 as indicated in the definition—try this by hand. The four resulting vectors form the
columns of the matrix AB. Matrix multiplication is also built into your calculator, enter A and B into your calculator
and compute A*B, do you get the same answer? Is it possible to multiply the matrices in reverse order, that is BA?
Discuss with your partner.
⎣
⎦
⎣
⎦
F =
1
3
⎡
⎣
⎢⎤
⎦
⎥
, G =
[]
24
, H =
2
3
1
1
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
, into your calculator. Compute the following. Note the sizes of the matrices that
you are multiplying and the size of the final answer. In particular concentrate on i, ii, vi, and vii. Do you get
identical answers for i? Discuss with your partner.
i. AB, BA ii. EB, BE iii. DC iv. AF
Problem 4. Use the random matrix command to construct random matrices of the sizes listed below. Write these
matrices down and compute the operations, and discuss as indicated.
i. A of size 3 × 5, and B of size 5 × 3. Compute AB and BA, and compare the answers. If your calculator
Problem 5. Another operation that is defined on matrices is “to take the transpose”. Read about the transpose
towards the end of section 2.1, including Example 8 and discuss it with your partner. The transpose operation, AT, is
built into your calculator. Compute the following, and write down your observations:
i. (AT)T ii. (C + D)T, CT + DT
iii. (rA)T, rAT iv. (AB)T, BTAT, ATBT
TI Project: LU Factorization
The problems in this project will illustrate a powerful method of solving systems of linear equations by
first factoring the matrix of coefficients into the product of two special matrices. Read Section 2.5 of the
text. Data for this project is in the LUFACT section of the PROJDATA program.
Programs. You will need the GAUSS,BGAUSS, and LU programs to speed up your operations.
GAUSS Zeros entries below the first nonzero entry of a matrix in a specified row. Prompts for a
matrix and a row, returns the modified matrix.
TI-83+ and TI-84+ (MP)
Operations.
[A]*[B] Matrix multiplication (2 > Y* 2 > Ze). A and B are compatible for
multiplication.
[A]-1 Computes the inverse of A (2 > Yie).
MATRIX: prompts for a matrix, enter A.
ROW: prompts for a row, enter 1.
to replace [J] with the matrix obtained by swapping rows 2 and 4 of [J]. Run the program GAUSS and
use [J] as the input. You can quickly access matrix [J] by going to the matrix menu and pressing the up
arrow, (as opposed to pressing the down arrow repeatedly to get to [J]).
MATRIX: prompts for a matrix, enter [J].
ROW: prompts for a row, enter 2.
−−
⎡
⎤
1230
TI-89
Operations.
aa*bb Matrix multiplication (aa p bb ¸). A and B are compatible for multiplication.
aa-1 Computes the inverse of A (aa Z · ¨ ¸).
LU A,L,U,P Computes the permuted LU factorization of A (2 I 4:Matrix scroll down to B:LU
and press ¸). Requires that you enter the four arguments.
matrix A =
−−
−
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
1230
24 65
11 13
25 101
run GAUSS, then wait for the prompts:
Matrix: prompts for a matrix, enter aa
Row: prompts for a row, enter 1
The calculator displays the matrix xx =
−−
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
1230
0005
0123
01 41
which is obtained from A by creating zeros in
Problem 1. For each one of the matrices below use the LU algorithm as in Example 2, Section 2.5, to compute the
LU factorization of A. Use the GAUSS program and your calculator to speed up the row operations. Write down the
matrices L and U, then verify that A = L*U.
13 530
11 221
−−
−−−
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
Problem 2. For the matrices A, B and C as above:
(a) Use the LU program in your calculator to find the permuted LU factorization. Write down the
Problem 3. For the matrix A as above and b =
2
3
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
. Follow the method used in Example 1, section 2.5 of the text to
solve the equation Ax = b.
Problem 4. For the matrix A as above and b =
2
3
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
. Follow the method used in Example 1, section 2.5 of the text to
solve the equation Ax = b, but now use the permuted factorization obtained in Problem 2(a).
(a) Explain why the equation PAx = Pb is equivalent to the two equations Ly = Pb and Ux = y.
(b) Solve Ly = Pb by forward substitution. Use your calculator to help with the arithmetic. Write
TI Project: Leontief Input-Output Model
The problems in this project will help you to appreciate the usefulness of linear algebra in real world
applications, such as, economic models. Review Section 2.6 of the text. Data for this project is in the
LEONTIEF section of the PROJDATA program.
Programs. You will need the CNORM, JCOL, and JROW programs to speed up your operations.
CNORM Computes the largest of the sums of the absolute values of the elements in each column.
Prompts for a matrix and delivers largest sum. (TI–83+ family only)
TI-83+ and TI-84+ (MP)
Operations.
[A]*[B] Matrix multiplication (2 > Y* 2 > Ze). A and B are compatible for
multiplication.
[A]-1 Computes the inverse of A (2 > Yie).
TI-89
Operations.
aa*bb Matrix multiplication (aa p bb e). A and B are compatible for multiplication.
aa-1 Computes the inverse of A (aa Z · ¨ ¸).
identity(k) Constructs a k×k identity matrix (2 I 4:Matrix scroll down to identity( and
press ¸).
Problem 1. Let C =
..…..
.……
…….
…….
3299 0565 0495 3636 0204 0483 0649
0089 0081 0333 0295 3412 0237 0020
1190 0901 0996 1260 1722 2368 3369
0063 0126 0196 0098 0064 0132 0012
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎦
⎥
⎥
⎥
⎥
⎥
⎥
and d =
33 501
23 527
263 985
6526
,
,
,
,
,
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎦
⎥
⎥
⎥
⎥
⎥
⎥
be the consumption
matrix and the demand vector, respectively, as in Exercise 14 of your text.
330 000
13 000
,
,
,
⎣
⎢
⎢
⎢
⎦
⎥
⎥
⎥
[x(5)] – C(5, 1, 5, 7)*x. Interpret your answer in terms of the economical model proposed. (Note: When
operating on matrices and vectors, your calculator is very picky about the difference between these and
their sizes; the product C(5, 1, 5, 7)*x yields a 1×1 vector which is subtracted from [x(5)], another 1×1
TI Project: Determinant and its Properties
In this project you will study determinants and how they relate to other topics covered in class. Some of
the calculations are to be done by hand. You will also create random matrices, and experiment with their
determinants to discover some properties on your own. Data for this project is in the DETPROP section of
the PROJDATA program.
Programs. You will need the RDIAG and RUTR programs to generate special random matrices.
TI-83+ and TI-84+ (MP)
Operations.
det([A]) Determinant of the matrix A (2 >(right arrow to MATH menu) e 2 > Ye).
[A]-1 Computes the inverse of A (2 > Yie).
rowswap(matrix,row1,row2)
Returns a matrix obtained from a given matrix by interchanging two rows. Enter the matrix, the
number of the first row to be interchanged, the number of the row with which it is to be
interchanged.
*row(value,matrix,row)
Returns a matrix obtained from a given matrix by multiplying all entries in a row by a nonzero
constant. Enter the value to multiply the row by, the matrix, the number of the row to be
TI-89
Operations.
det(aa) Determinant of the matrix A (2 I 4:Matrix 2:det( ¸).
TI Project: Determinant and its Properties 2
aaT Computes the transpose of A (2 I 4:Matrix ¸).
randMat(r,c) Returns as random matrix of size r × c with integer entries in the interval [–9, 9]. (2
I 4:Matrix scroll down to randMat( and press ¸) Requires two arguments—
row size and column size of the desired matrix—close parenthesis and press ¸.
Problem 1. Read the definition of determinant that is given in section 3.1 of your text. Use it to compute
⎣
⎦
Problem 2. Read Theorem 1, and Examples 2 and 3 from this section, discuss them with your partner. Use
2136
⎡
⎤
Problem 3. Read the numerical note at the end of this section. It contains useful information, about numerical
computations.
Problem 4. Use the program RDIAG to generate random diagonal matrices of sizes 3, 4, and 5; save the
matrices in memory. Use the det command to compute the determinant of the matrices. Write down your
observations. Describe any relation between the diagonal entries in the matrices and their determinants?
Problem 5. Use the program RUTR to generate random upper triangular matrices of sizes 3, 4, and 5; save the
Problem 6. This problem will help you to understand how the determinant of a matrix is affected when
elementary row operations are performed on the matrix.
a. Use the random matrix command to generate a random matrix of size 3, and one of size 4. Store the
matrices in memory. Write these matrices down.
d. Use the det command to compute the determinants of the matrices that you created in c. Write these
determinants down. Compare with your answers in b. Write down your observations.
e. Use the random matrix command to generate a random matrix of size 4, and one of size 5. Store the
matrices in memory. Write these matrices down.
f. Use the det command to compute the determinant of these matrices, write these determinants down.
g. Use row swap operation of your choice—different ones—on each of these two matrices. Store the new
matrices in memory and them down.
h. Use the det command to compute the determinants of the matrices that you computed in g. Write them
down.
i. Write down your observations. What happens if you do two swap operations on the same matrix? Three
swap operations? Four? Explain.
Problem 7. This problem will help you to understand how the determinant of a matrix relates to matrix
operations.
a. Use the random matrix command to generate two random matrices of size 5. Store the matrices in memory
as A and B. Write these matrices down.
Problem 8. This exercise will reveal the importance of the determinant in establishing invertibility of a matrix.
a. Use the random matrix command to generate two random matrices of size 4. Store the matrices in memory
as A and B. Write these matrices down.
b. Use the det command to compute the determinant of each of the matrices A and B, write these determinants