Math 141
Chapter 2 – Matrices, System of Equations
TAMU – Summer 2014
– A Matrix is a rectangular array of numbers. It consists of rows and columns
– The order of a matrix with m rows and n columns is given by m x n and is read as 'm by n'.
– Examples of Matrices
– Every element in the matrix is referred to by it's row and column number. That is, the element in the
ith row and jth column of a matrix A will be ___________.
– A Row matrix is a matrix with ONE ROW and any number of columns.
– A Column matrix is a matrix with ONE COLUMN and any number of rows.
– A Square matrix is one where the number of rows is the same as the number of columns.
– Two matrices are equal if they have the same order and all corresponding entries are equal.
– Multiplication of a Matrix by a Number (or a scalar) : If c is a number and A is a matrix, then cA
is a matrix obtained by multiplying every entry in A by c.
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Math 141
Chapter 2 – Matrices, System of Equations
TAMU – Summer 2014
– Transpose of a Matrix : is obtained by interchanging the rows and columns. The transpose of matrix
A is represented as AT.
– Addition and Subtraction of Matrices : is possible ONLY IF THE MATRICES ARE OF THE
SAME ORDER.
These operations are performed by adding or subtracting corresponding entries in the matrices.
– The Zero Matrix is a matrix of any order, such that all its entries are ZERO. It's given by O.
If A be any matrix, such that the order of A is the same as that of O, then
1) A + O = O + A
2) A – A = O
– The Identity Matrix is a SQUARE MATRIX where the diagonal elements are 1, and all the other
entries are zero.
If A be a square matrix of order n, and I be an identity matrix of order n, then
AI=IA=A
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Math 141
Chapter 2 – Matrices, System of Equations
TAMU – Summer 2014
1. Given the matrices A, B, C, and D, perform the following operations and write the final matrices:
A=
1
2
–1
B= 4
0
5
3
10
7
–1
6
3
C= 1
3
5
D= 2
0
–1
a) 2A – B
b) (2A – B )T
c) C T - D / 2
d) Is A + B = B + A ?
e) Is 2(A + B) = 2B + 2A
2. Find x, y, z if
1
2
–1
x–1
3
1
=
3
0
6
2+x
–1
z
1
– 10
3
+
2
y–4
3
3. If X is a 3 x 5 matrix, Y is a 5 x 5 matrix, Z is a 5 x 3 matrix, and U is a 2 x 2 matrix, then state the
order of the following matrices (if the operation is possible).
a) X + Z
3
b) X + Z T
c) 2U
d) Y + Y + Y
e) 10Z
Math 141
Chapter 2 – Matrices, System of Equations
TAMU – Summer 2014
Matrix Applications
Calculator:
MATRIX (2nd x^-1) → Scroll to the matrix → Scroll to MATH → Scroll to B: rref → MATRIX (2nd
x^-1) → NAMES → Scroll to the matrix you want to work with → ENTER
1. Using the rref calculator function for matrices, check whether the following equations have a unique
solution, no solution, or infinitely many solutions. If (a) solution/solutions exist, find them.
a) 2x + 3y + 4z = 5
x – 2y + 2z = 4
x + y + z = -2
b) x – y +z = 1
2x + 3y – 2z = 1
3x + 2y – z = 1
d) 3x – 2y + 4z = 21
2x + y - 2z = 7
x + 4y – 8z = -7
e) -2x + 6y + 4z = 12
3x – 9y – 6z = -18
f) x + 2y + z – u = -2
x + 2y + 2z + 2u = 9
y + z – u = -2
y – 2z + 3u = 4
g) x + 2z + 3u = 4
y + 2z + 3u = 5
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Math 141
Chapter 2 – Matrices, System of Equations
TAMU – Summer 2014
2. A furniture company makes chairs, footstools and couches out of fabric, wood and stuffing. The
number of units of each of these materials needed for each of the products is given in the table. How
many of each product can be made if there are 54 units of wood, 63 units of fabric, and 43 units of
stuffing available?
Wood
Fabric
Stuffing
Chair
2
2
1
Footstool
3
1
1
Couch
1
3
3
3. Every software developer has to write three types of documents for every new product he develops–
a technical document for other developers to understand how the program works, a business document
for the marketing group, and a customer-friendly document for the consumer group. Each document
has to go through three managers. The technical document takes 2 hours to be proof-read by the first
manager, 3 hours by the second manager, and 3 hours by the third.. The business document takes 4
hours to be read by the first manager, 2 hours by the second, and 3 hour by the third. The
customer-friendly document takes 2 hours to be read by the first manager, 4 hours by the second, and 3
hours by the third. If the first manager has 34 hours, the second manager has 35 hours, and the third
manager has 36 hours for this job, how many documents of each type can they process?
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Math 141
Chapter 2 – Matrices, System of Equations
TAMU – Summer 2014
4. A store sells a total of 300 shirts, trousers and dresses. The sales of the dresses equaled sum of the
other two. The dresses cost $30, the shirts cost $40, and the trousers cost $50, each. If the store sold
$11,500 worth of these apparels in that month, how many of each were sold?
5. After trick-or-treating, Ted and Rob counted their candies. Ted had five less than twice as many
candies as Rob (sob sob !). If there are a total of 40 candies, how many candies do each of them have?
6. A store sells a total of 300 sandals in a certain month. The sandals come in three styles – pumps,
slides, T-strap. The sales of the slides equaled the other two. The pumps cost $30, slides cost $40, and
T-straps cost $50. If the store sold $11,500 worth of these sandals in that month, how many sandals of
each style were sold?
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Math 141
Chapter 2 – Matrices, System of Equations
TAMU – Summer 2014
7. You and your friends are making cakes for Halloween. You make three kinds of cakes – Batman,
Witch, and Angel. The Batman cake uses 2 cups of butter and 2 cups of sugar. The Witch cake uses
1cup of butter and 2 cups of sugar. The Angel cake uses 7 cups of butter and 4 cups of sugar. You
have got 30 cups of butter and 30 cups of sugar available at home.
a) How many of each type of cake can you and your friends make?
b) List all possible combinations of the number of cakes of each type.
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Math 141
Chapter 2 – Matrices, System of Equations
TAMU – Summer 2014
Matrix Multiplication
Given two matrices – A of size m x p, and B of size q x n, their product AB = C is defined if p = q, and
C is a matrix of order m x n.
If
A= 1
2
3
4
5
6
a) compute AB =
and B =
–1
8
0
9
–2
– 10
b) compute BA
Is AB = BA ?
NOTE : If AB is defined, it's NOT necessary that BA will also be defined.
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Math 141
Chapter 2 – Matrices, System of Equations
6. Find x, y and z if it's given that
– 9/4
1
5
12
x–1
3
7
1
2
y+2
–4
0
3
z– 2
0
1
=
9
TAMU – Summer 2014
Math 141
Chapter 2 – Matrices, System of Equations
TAMU – Summer 2014
7. Matrix M shows the nutritional content of 3 different types of dog food. A dog is fed 27 grams of
kibbles, 55 grams of bits, and 68 grams of chunks.
a) Write the column matrix N representing the amount, in grams, of each type of food fed to a dog.
b) Find the product MN. What does the product represent?
c) How many units of all the vitamins combined, does the dog get in a meal?
8. Matrix A shows the number of calories from fat, protein, and carbohydrates per unit of each food.
Matrix B represents the number of units of food eaten by a person.
a) Find AB.
b) What does AB represent?
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