ME 404
ADVANCE MATHEMATICS
FOR ME
Engr. Zedric Rei C. Canicon
Lecturer
©2023
1
Batangas State University
Matrices: Review,
Operations & Vectors
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Matrix
Matrix is a rectangular array of real numbers arranged in “m” rows and “n”
columns. The term “matrix” was introduced by the English mathematician James
Joseph Sylvester (1814- 1897) in 1850.
The size of a matrix is determined by the number of rows and columns. The
expression "m x n" is the dimension or order of the matrix. If the matrix has only one
column, it is called a column matrix and if it has only one row, it is called a row
matrix. The following is a 3 x 3 matrix or square matrix (i.e. 3 rows and 3 columns).
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Matrix
A matrix is called by m x n matrix where m is designated as row and n as
column (note: rows always come first). The size of matrix is determined by the m x n
matrix.
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Matrix
A matrix is a rectangular array of numbers or functions enclosed in brackets.
The numbers (or functions) are called entries or, less commonly, elements of the
matrix.
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Matrix
1
23
55 −6
1/2 3
4
12
−7
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Matrix
1
23
55 −6
1/2 3
4
12
−7
SQUARE MATRIX
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Matrix
1
23
55 −6
1/2 3
4
12
−7
LEADING ENTRY
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Matrix
1
23
55 −6
1/2 3
4
12
−7
DIAGONAL ENTRIES
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Matrix
1
0
55 −6
1/2 3
0
0
−7
LOWER TRIANGULAR
MATRIX
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Matrix
1 23
0 −6
0 0
4
12
−7
UPPER TRIANGULAR
MATRIX
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Vectors
A vector is a matrix with only one row or column. Its entries are called the
components of the vector. A row vector is of the form
A column vector is of the form
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Matrix
1 0
0 −6
0 0
0
0
−7
DIAGONAL MATRIX
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Operations on Matrix
Addition/ Subtraction of Matrices
Matrices having the same number of rows and columns are added by adding
corresponding elements. Thus, for all m x n matrices A and B as example, their sum 𝐴 +
𝐵 is a matrix 𝐶 = 𝑐𝑖𝑗 such that 𝑎𝑖𝑗 + 𝑏𝑖𝑗 = 𝑐𝑖𝑗 , for 𝑖 = 1,2, … , 𝑚 and 𝑗 = 1,2, … , 𝑛.
Since the sum of A and B is itself an m x n matrix, it is said that the set of all m x n
matrices is closed under matrix addition. Addition of matrices of different sizes is undefined.
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Operations on Matrix
Example 1:
If 𝐴 =
−4
0
6 3
5
and 𝐵 =
1 2
3
−1 0
, what is the value of 𝐴 + 𝐵 ?
1 0
Solution:
𝐴+𝐵 =
−4 + 5 6 + (−1) 3 + 0
0+3
1+1
2+0
𝟏
𝟓
𝟑
𝑨+𝑩=
𝟑 𝟐 𝟐
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Operations on Matrix
Example 2:
16 21 −3
5 −1 0
If 𝐶 = 8 −10 6 and 𝐷 = 3 1
0 , what is the value of 𝐶 + 𝐷 ?
16 10
2
8 −7 −7
Solution:
16 + 5 21 + −1
−3 + 0
𝐶+𝐷 = 8 + 3
−10 + 1
6+0
16 + 8 10 + (−7) 2 + (−7)
𝟐𝟏 𝟐𝟎 −𝟑
𝑪 + 𝑫 = 𝟏𝟏 −𝟗 𝟔
𝟐𝟒 𝟑 −𝟓
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Operations on Matrix
Example 3:
−4 6
If 𝐴 =
0 1
5
3
and 𝐷 = 3
2
8
−1 0
1
0 , what is the value of 𝐴 + 𝐷 ?
−7 −7
Solution:
Matrices to be solved do not have the same order/size of matrix that is why
the operation cannot be performed.
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Operations on Matrix
Scalar Multiplication of Matrices
The product of any 𝑚 𝑥 𝑛 matrix A = 𝑎𝑖𝑗 and any scalar c (number c) is
written cA and is the m x n matrix cA = 𝑐𝑎𝑖𝑗 obtained by multiplying each entry of
A by c.
-1A is simply written as –A and is called the negative of A. Similarly, (-k)A is
written as –kA. Also, A+(-B) is written as A-B and is called the difference of A and
B and must have the same size.
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Operations on Matrix
Example 3:
2.7 −1.8
If 𝐴 = 0
0.9 , find the value of the following:
9.0 −4.5
a. -A
b.
10
𝐴
9
c. 0A
Solution:
2.7 −1.8
−2.7 1.8
a. 𝐴 = 0
0.9 then− 𝐴 = 0
−0.9
9.0 −4.5
−9.0 4.5
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Operations on Matrix
Example 3:
2.7 −1.8
If 𝐴 = 0
0.9 , find the value of the following:
9.0 −4.5
a. -A
b.
10
𝐴
9
c. 0A
Solution:
2.7 −1.8
10
b. 𝐴 = 0
then
𝐴=
0.9
9
9.0 −4.5
10
× 2.7
9
10
×0
9
10
× 9.0
9
10
× −1.8
9
10
𝟏𝟎
× 0.9 . 𝑨
9
𝟗
10
× −4.5
9
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𝟑 −𝟐
= 𝟎 𝟏
𝟏𝟎 −𝟓
Operations on Matrix
Example 3:
2.7 −1.8
If 𝐴 = 0
0.9 , find the value of the following:
9.0 −4.5
a. -A
b.
10
𝐴
9
Solution:
2.7 −1.8
𝟎
a. 𝐴 = 0
0.9 then 𝟎 𝑨 = 𝟎
9.0 −4.5
𝟎
c. 0A
𝟎
𝟎
𝟎
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Operations on Matrix
Rules for Matrix Addition and Scalar Multiplication
From the familiar laws for the addition of numbers, we obtain similar laws for
the addition of matrices of the same size m x n, namely,
Here 0 denotes the zero matrix of (size m x n), that is, the m x n matrix with
all entries zero. If m=1 or n=1, this a vector, called a zero vector.
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Operations on Matrix
Hence matrix addition is commutative and associative. Similarly, for scalar
multiplication we obtain rules
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Operations on Matrix
Matrix Multiplication
The product C = AB (in this order) of an m x n matrix A = 𝑎𝑗𝑘 multiplied by
an r x p matrix B = 𝑏𝑗𝑘 is defined if and only if r = n and is then the m x p matrix C =
𝑐𝑗𝑘 with entries
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Engr. Zedric Rei C. Canicon
Operations on Matrix
Matrix Multiplication
The condition r = n means that the second factor, B, must have as many rows
as the first factor has columns, namely n. A diagram of sizes that shows when
matrix multiplication is possible is as follows:
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Engr. Zedric Rei C. Canicon
Operations on Matrix
The entry 𝑐𝑗𝑘 is obtained by multiplying each entry in the jth row of A by the
corresponding entry in the kth column of B and then adding these n products.
For instance, 𝑐21 = 𝑎21 𝑏11 + 𝑎22 𝑏11 + ⋯ + 𝑎2𝑛 𝑏𝑛1 , and so on. One calls this
briefly a multiplication of rows into columns. For n = 3, this is illustrated by
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Operations on Matrix
Example 1: Multiplication of Matrix and Vector
Find Matrix AB and BA.
4 2
𝐴=
1 8
3
𝐵=
5
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Engr. Zedric Rei C. Canicon
Operations on Matrix
Example 2:
Find Matrix AB as A x B = AB
3
5 −1
𝐴= 4
0
2
−6 −3 2
2 −2 3
𝐵= 5 0 7
9 −4 1
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1
8
1
Operations on Matrix
Matrix Multiplication Rules
provided A, B, and C are such that the expressions on the left are
defined; here, k is any scalar. (b) is called the associative law. (c) and (d)
are called the distributive laws.
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Operations on Matrix
Transpose Matrix
If matrix A is reflected in its main diagonal, so that all rows become
columns and all columns become rows without changing their relative
order of entries in the rows and columns, the result is a transpose
matrix, AT.
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Operations on Matrix
Example 1:
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Operations on Matrix
Example 2:
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Operations on Matrix
Example 3:
Find the transpose of Matrix Z.
5
16
33 21
𝑍=
−2 −50
8
0.5
15
4
0
−4/5
Solution:
5
𝑍 𝑇 = 16
15
33
21
4
−2
8
−50
0.5
0
−4/5
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Operations on Matrix
Rules for Transposition:
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Operations on Matrix
Example: Application of Matrix Multiplication
Supercomp Ltd produces two computer models PC1086 and PC1186. The matrix A shows
the cost per computer (in thousands of dollars) and B the production figures for the year 2010 (in
multiples of 10,000 units.) Find a matrix C that shows the shareholders the cost per quarter (in
millions of dollars) for raw material, labor, and miscellaneous.
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Operations on Matrix
Special Matrices
Symmetric matrices are square matrices whose transpose equals the
matrix itself. Skew-symmetric matrices are square matrices whose
transpose equals minus the matrix.
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Engr. Zedric Rei C. Canicon
Operations on Matrix
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Engr. Zedric Rei C. Canicon
Operations on Matrix
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Operations on Matrix
Special Matrices
Scalar matrix is a diagonal matrix whose elements main diagonal are
the same.
3 0
0 3
0 0
0
0
3
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Engr. Zedric Rei C. Canicon
Operations on Matrix
Special Matrices
Identity matrix is a diagonal matrix whose elements main diagonal are
all 1.
1 0
0 1
0 0
0
0
1
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Operations on Matrix
Special Matrices
Trace of matrix is the sum of the entries of main diagonal of a square
matrix.
1
23 4
55 −6 12
1/2 3 −7
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Engr. Zedric Rei C. Canicon
Operations on Matrix
Given with matrices, solve the following operations.
1
𝑍 = 55
1/2
1
𝐸=
3
a. 𝑍𝐸
23 4
−6 12
3 −7
7 0
5 0
7
0
5
0
𝐷=
−4
−7
−2 −1/2
15 3 1 12
𝑌=
3 −8 0 1
b. 3𝐷 − 2 𝑌 𝑇
c.𝐸𝑍 𝑇
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