Symmetric Matrix

What is Symmetric Matrix?
A square matrix is said to be a symmetric matrix when the matrix and its transpose are equal.
Symbolically, it can be represented as A = AT.  Transpose means that the rows are written as columns and columns are written as rows.

Inverse of a Symmetric Matrix
Let us see how inverse of a symmetric matrix is calculated in two special cases namely diagonal matrix and 2x2 matrixes.

Inverse of a Diagonal Matrix
Diagonal matrix is one of the special cases of symmetric matrix. The matrix elements present in positions other than the main diagonal will be zero in the diagonal matrix.  By replacing every element in the main diagonal of the diagonal matrix with its corresponding reciprocal, the inverse of a diagonal symmetric matrix is obtained.

The equation XX-1  =  X-1X = I confirms that X-1  is the inverse of X.  In this expression, I represent identity matrix.

Your professor might ask you to find inverse of a diagonal matrix with one of the elements in the main diagonal as zero. Be careful, it’s a tricky question. If any of the diagonal elements is zero, then the inverse of that diagonal symmetric matrix cannot be calculated i.e., it has no inverse.

Then the determinant, denoted by |X| is calculated by the formula
|X| = X11X22 – X12X21

Then, the inverse of symmetric matrix is given by

Determinant of Symmetric Matrix
The determinant of symmetric matrix is n (n+1)/2 scalars, which represents all the entries in the matrix that are above the main diagonal and that are present on the main diagonal.

Properties of Symmetric Matrices

Symmetric Matrix properties are:

• If X and Y are two symmetric matrices of size m x m, then
o X+Y is also a symmetric matrix as (X+Y)T = XT+YT = (X+Y).
o XY is not symmetric as (XY)T = YTXT = YX which is not equal to XY.

• If the given matrix is a diagonal matrix, then it is a symmetric matrix.

• The product of a symmetric matrix and its transpose is a symmetric matrix.

Math Proportions

To understand what a proportion means, we should first know what are ratios? A ratio is a comparison of two quantities by division.
Definition of proportion (Proportion definition)
A proportion is when you set two ratios equal to each other. Two ratios when are set equal to each other are called in proportion. So this is how we define proportion. A proportion can be written in two ways: -
• In fractions like a/b = c/d
• Using a colon a:b = c:d
For example: -
5/4 = 25/20
Or 5:4 = 25:20
How to solve proportions? (Solving proportions or proportion solver)
Proportions can be solved by cross – multiplication. When the terms of a proportion are cross multiplied, the cross products are equal. So when the two ratios are equal their cross products are equal.
For example if a/b and c/d are in proportion then ad = bc
To solve the missing value in a proportion, we simply cross multiply and simplify.
For instance if we have been given that x/4 = 1/12
To solve proportion, we cross multiply first, we get
12x = 4
Hence, x = 3
This is how we solve proportions.
Direct proportion
Direct proportion is defined as the increase in one quantity due to the increase in the other quantity. When two quantities are in direct proportion, their ratios are constant. For example: -
If a and b are in direct proportion then if there will be any increase in ‘a’ then it would cause the changes in ‘b’ too by the same factor. We can represent the directly proportional terms by the symbol ‘a’
For example: -
If x is directly proportional to y, then we can write it as: -
x a y
or x = ky
where k is called the constant of proportionality.

Greatest Common Factor & Word Problems

Greatest Common Factor Definition: 
Greatest common factor, in short GCF is the highest of the common factors of two or more numbers

Greatest Common Factor

For example:
Find the GCF of 12, 24 and 18
Solution: First we find the factors of,
12 = 1,2,3,4,6,12
24 = 1,2,3,4,6,8,12,24
18 = 1,2,3,6,9,18
From the above list of factors, we can see that 6 is the highest common factor
 So, the GCF of the number 12, 24 and 18 is 6

Let us solve some Greatest common factor word problems (GCF word problems):
1. Edina has 72 inches and 90 inches wide cloth strips. How wide should she cut them into strips of equal width that are as wide as possible?
Solution: To find the greatest possible width, we need to find the common factors of 72 and 90
Factors of,
72 = 1,2,3,4,6,8,9,12,18,24,36 and 72
90 = 1,2,3,4,5,6,9,10,15,18,30,45 and 90
Common factors are 1, 2, 3, 4, 6,9,18
Highest common factor is 18
Edina needs to cut each strip 18 inches wide

2. Wilma is making flower arrangements. She has 8 roses and 16 daisies. If she wants to make arrangements identical with no flower left over, what is the greatest number of arrangements possible?
Solution: Factors of 8 and 16 are
8 = 2 x 2 x 2
16 = 2 x 2 x 2 x 2
Common factors are 2, 2, and 2
Since 2 is repeating thrice, the GCF will be product of them, 2 x 2 x 2 = 8
Wilma can make 8 arrangements

GCF learning (GCF learn free):
GCF is Greatest Common Factor, which is the highest of the common factors of the given numbers.
Common Factors are factors which are common to the list of factors of the given numbers
Example: Find the GCF of 12 and 16
Solution: factors of 12 are 1, 2, 3, 4, 6 and 12
 Factors of 16 are 1, 2, 4, 8 and 16
  common factors are 1, 2 and 4
GCF of 12 and 16 will be 4, it being the highest common factor