Chapter 1
Section 1.5
Matrix Operations
Matrices
A matrix (despite the glamour of the movie) is a collection of numbers arranged in a
rectangle or an array. We use variables like A, B, C, …, [capital letter] to stand for a
matrix.
We use what are called double scripted variables with a lower case letter of the
matrix to refer to the entries in a matrix. The numbers in the subscript give the
position the variable is located in with the first number referring to the row and the
second number the column. The dimensions or order of the matrix is given in the
form (number of rows)  (number of columns). Don't multiply leave it this way!
4 3  1  a11 a12
0 7 1    a
 21 a22
2 

a13 
a23 
a11  4
a12  3
a13  1
a21  0
a22  7
a23 
1
2
We name this matrix A
This matrix has 2 rows and 3
columns, or has order or dimension
2  3 "read 2 by 3".
b11 b12 b13  5 7  3
b
  9 12 0 
b
b
 21 22 23  

b31 b32 b33  1 4 6 
What do we name this?
B
What is the entry b21?
9
What is the entry b12?
7
What is the variable for 4?
b32
What are the dimensions?
33
Matrix Operations
Adding & Subtracting Matrices
The way that matrices are added or subtracted is to add or subtract their
corresponding entries. This means that the matrices must be of the same dimensions
or order. If they are not we say the two matrices are not the same dimensions we say
the matrices are nonconformable.
4   35
(6)  4  8  2
 3  6  5
A  B   1 9    3  5   1  3
9  (5)   4 4 
12  4  7 10  12  (7) (4)  10 5 6 
The matrices C and D are
nonconformable. They can not
be added even though they both
have 6 entries.
4 7 
C  D  12  3  5 6 1 0  2 6
Matrix C is 2  3
 9 1 
Matrix D is 1  6
Multiplication by a Scalar
We can multiply a matrix by a number (sometimes called a scalar) by multiplying
each entry in the matrix by the number. This operation can always be done. We say
it is always conformable.
 4   2(4)   8 
2A  2 3  2(3)   6
 7   2(7)   14 
1
3
6 2  1  13 (6)
B 
   1 (7)
7
9
12

 3
1
3
1
3
1
3
(2)
(9)
1
3
1
3
(1) 2 23

(12)   73 3

4 
1
3
We can begin to combine more than one operation at a time.
3C  4D  35 2  4 2 1  3(5) 3(2)  4(2) 4(1)  15 6   8 4  7 10
 4
28
3E  7 F  34  2 6  7 3  12  6 18   21
1
 7 
What you get here is
nonconformable since
the first matrix is 1  3
and the second matrix is
3  1.
Vectors in ℝ𝑛
A vector (or column vector) is a matrix with
only 1 column. If number of rows (entries
or components) is n we call the vector an
n-dimensional vector and it is in the set ℝ𝑛 ,
which consists of all vectors with n rows.
𝑥1
x = ⋮ x is a vector in ℝ𝑛
𝑥𝑛
𝑥1
ℝ𝑛 = ⋮ : 𝑥1 , ⋯ 𝑥𝑛 ∈ ℝ
𝑥𝑛
Linear Combinations of Vectors
If each entry in a vector is a linear combination of the variables 𝑥1 , 𝑥2 , ⋯ , 𝑥𝑛 then that
vector can be rewritten (expressed) as each variable being multiplied by a vector. We call
this a linear combination of vectors.
𝑎11
𝑎12
𝑎1𝑛
𝑎11 𝑥1 + 𝑎12 𝑥2 + ⋯ + 𝑎1𝑛 𝑥𝑛
𝑎2𝑛
𝑎21
𝑎22
𝑎21 𝑥1 + 𝑎22 𝑥2 + ⋯ + 𝑎2𝑛 𝑥𝑛
=
𝑥
+
𝑥
+
⋯
+
𝑥
1
2
𝑛
⋮
⋮
⋮
⋮
𝑎𝑚1
𝑎𝑚2
𝑎𝑚𝑛
𝑎𝑚1 𝑥1 + 𝑎𝑚2 𝑥2 + ⋯ + 𝑎𝑚𝑛 𝑥𝑛
2𝑥1 − 𝑥3
2
−1
0
𝑥1 + 5𝑥2 − 7𝑥3 = 𝑥1 1 + 𝑥2 5 + 𝑥3 −7
𝑥2
0
0
1
2𝑥4 − 5𝑥6
2
−5
3𝑥6
= 𝑥4 0 + 𝑥6 1
−2
1
−2𝑥4 + 𝑥6
9
0
9𝑥4
Vector Form of Solution
The general solution to a linear system of equation can be expressed as a linear
combination of vectors. The number of vectors required to do this is the same as the
number of independent variables and a vector for the constants (if required for a
nonhomogeneous system). Some examples are given below.
Augmented Matrix
1
0
0
0
0
0
0
0
0 −5
1 3
0 0
0 0
1
0
0
0
2
0
0
0
0
1
0
0
0
−1
0
0
3
−2
0
0
0
0
1
0
0
0
0
0
Vector Form
General Solution
3
7
2
0
𝒙𝟏 = 𝟑 + 𝟓𝒙𝟑
𝒙𝟐 = 𝟕 − 𝟑𝒙𝟑 + 𝒙𝟒
𝒙𝟓 = 𝟐
2
−5
0
0
3 + 5𝑥3
3
7
7 − 3𝑥3 + 𝑥4
= 0 + 𝑥3
𝑥3
0
𝑥4
2
2
5
−3
1 + 𝑥4
0
0
0
1
0
1
0
𝑥2 = 2 − 2𝑥3 − 3𝑥5
𝑥4 = −5 + 2𝑥5
𝑥1
0
2 − 2𝑥3 − 3𝑥5
2
𝑥3
0 +𝑥
=
1
−5 + 2𝑥5
−5
𝑥5
0
𝑥6
0
1
0
0 +𝑥
3
0
0
0
0
−2
1 +𝑥
5
0
0
0
0
−3
0 +𝑥
6
2
1
0
0
0
0
0
0
1
Multiplying Matrices
This is not as obvious an operation as you might think!
It is not as easy as addition or subtraction that you get
with the corresponding entries!
What you do is to multiply each entry in a row on the matrix on the left with its
corresponding entry in a column of the matrix on the right and add them up.
AB = (rows of matrix A) (columns of matrix B)
Look at the example below:
5
2 1 7    2(5)  1(9)  7(6)   10  9  42   23
AB  
9 







4
5
3
4
(
5
)

5
(
9
)

3
(

6
)
20

45

18
47

   6 
 
 

 
23
31
The dimensions of the result are given by
the rows of A and columns of B.
21
The matrix A is 2  3 and the matrix B is 3  1. The number of columns for the
matrix on the right must be the same as the number of rows for the matrix on the
left or else they are nonconformable!
6 0  2
 4(6)  1(3) 4(0)  1(5) 4(2)  1(1)  27 5  7

3 5 1 



CD  4 1 
12
13
23
5 3 2 6 0  2
ED  
 3 5 1 
0
4
1



23
23
The matrix E and the matrix D are
nonconformable even though they are the
same dimensions. The columns and rows do
not match up!
3 2  5  3(5)  2(2)  19
FB  

 




1  2 2 1(5)  2(2)  1 
If you multiply a 2  2 matrix by a 2  1 matrix you get another 2  1 matrix!
Matrix Multiplication and Linear Systems
A linear system of equations can be written (expressed) as a matrix equation of the form
𝐴x = b where A is the coefficient matrix, x is a vector consisting of the variables
𝑥1 , 𝑥2 , ⋯ 𝑥𝑛 and b is a vector of the constants.
𝑎11
𝑎21
𝐴x = ⋮
𝑎𝑚1
𝑎12
𝑎22
⋮
𝑎𝑚2
⋯
⋯
⋯
𝑎1𝑛
𝑎2𝑛
⋮
𝑎𝑚𝑛
𝑥1
𝑎11 𝑥1 + 𝑎12 𝑥2 + ⋯ + 𝑎1𝑛 𝑥𝑛
𝑏1
𝑥2
𝑎21 𝑥1 + 𝑎22 𝑥2 + ⋯ + 𝑎2𝑛 𝑥𝑛
𝑏2
=
=
=b
⋮
⋮
⋮
𝑥𝑛
𝑎𝑚1 𝑥1 + 𝑎𝑚2 𝑥2 + ⋯ + 𝑎𝑚𝑛 𝑥𝑛
𝑏3
2𝑥1 + 𝑥3 − 3𝑥4 = 8
𝑥1 + 5𝑥2 + 𝑥4 = 23
𝑥1 + 𝑥2 − 𝑥3 = 5
Matrix Column Vectors
The columns of a matrix can be
expressed as vectors. The notation 𝐴𝑖
refers to the ith column of matrix A.
3
𝐴= 1
4
0
0
3
6 , 𝐴1 = 1 , 𝐴2 = 6
5
5
4
2
1
1
𝑥
8
0 1 −3 𝑥1
2
5 0
4 𝑥3 = 23
5
1 −1 −1 𝑥4
𝑎11
𝐴= ⋮
𝑎𝑚1
𝐵=
⋯ 𝑎1𝑛
𝑎11
𝑎1𝑛
⋱
⋮ , 𝐴1 = ⋮ , ⋯ , 𝐴𝑛 = ⋮
⋯ 𝑎𝑚𝑛
𝑎𝑚1
𝑎𝑚𝑛
𝐴 = 𝐴1 , 𝐴2 , ⋯ , 𝐴𝑖 , ⋯ , 𝐴𝑛
4
2
7
8
9
4
7
9
, 𝐵1 =
, 𝐵2 =
, 𝐵3 =
1
2
8
1
Column Vectors and Linear Systems
A linear system of equations written in
matrix form as 𝐴x = b can be expressed
as linear combination of the column
vectors as shown to the right.
Column Vectors and Matrix Multiplication
The product of a 𝑚 × 𝑝 matrix A and a
𝑝 × 𝑛 matrix B can be formed by taking
each column vector of B and multiplying it
by the matrix A.
3
𝐴=
4
1 3
−1
, 𝐵 = −2 0 , 𝐵1 =
1
5 1
1
3 0 −1
𝐴𝐵1 =
−2 =
4 2 1
5
3
3 0 −1
𝐴𝐵2 =
0 =
4 2 1
1
0
2
𝑎11
𝑎12
𝑎1𝑛
𝑏1
𝑥1 ⋮ + 𝑥2 ⋮ + ⋯ + 𝑥𝑛 ⋮ = ⋮
𝑎𝑚1
𝑎𝑚2
𝑎𝑚𝑛
𝑏𝑚
𝐴𝐵 = 𝐴𝐵1 , 𝐴𝐵2 , ⋯ , 𝐴𝐵𝑛
𝑏11
𝑏12
𝑏1𝑛
= 𝐴 ⋮ ,𝐴 ⋮ ,⋯,𝐴 ⋮
𝑏𝑝1
𝑏𝑝2
𝑏𝑝𝑛
1
3
−2 , 𝐵2 = 0
5
1
−2
5
8
13
𝐴𝐵
1 3
3 0 −1
=
−2 0
4 2 1
5 1
−2 8
=
5 13