Diagonal Matrix Properties

A (mxm) matrix is a square array of numbers which has equal number of rows and columns here denoted by ‘m’. The elements of a square matrix that start from the left corner towards the right corner like a diagonal is called the Main Diagonal of a Matrix. For example, a 3x3 matrix given by
A=[-1  3  4]
 [2   1  0]
 [7  0   5]
The elements [ -1,1,5] form the elements of the main diagonal. Now consider A=(aij) to be a square matrix then the elements in aii is the diagonal of a matrix.  A square matrix in which all elements except the elements of the leading diagonal are zero is called a Matrix Diagonal. In this all the non diagonal elements are zero. For instance, a square matrix given by A=[aij]nxn to be a diagonal-matrix , aij is equal to zero wherever ‘i’ is not equal to ‘j’. Example of a diagonal-matrix can be given as follows, a 3x3 matrix given by
A=[2    0    0]
     [0   -1    0]
     [0    0    5]

Let us now take a look at the Diagonal Matrix Properties:
• If A and B are two diagonal matrices of the same order then, A+B and AB are also diagonal matrices.  Also AB=BA, that is they commute
• A square matrix is diagonal only if it is triangular and normal
• If A is a diagonal-matrix then the adjugate of A is also a diagonal matrix
• Any diagonal matrix is symmetric
• It is a scalar matrix when all the entries of the diagonal are equal
Determinant Diagonal Matrix
The determinant of a 2x2 square matrix A=[a    b] can be defined as |A|=ad-bc. It is denoted by det(A)
    [c    d]
Let A=[aij] be a nxn matrix where n is greater than or equal to two. We can define determinant,
|A|= a11C11 + a12C12+……..+a1nC1n which is equal to summation(i= 1 to n)[a1iC1i]. If A is a nxn diagonal matrix then |A| is the product of the elements in its main diagonal.
Let A = [a11       0       0……     0]
             [0         a22     0 ……   0]
     [0          0      a33……    0]
     [..         ..      ..     ..]
     [..         ..      ..     ..]
     [0          0        0…. .ann]

be a nxn diagonal matrix, then the determinant of this matrix can be given by a simple formula, det (A)=|A|= pi(i= 1 to n)[aii]
Consider the following example,
A=[2   0    0]
      [0   5    0]
      [0   0    1]

then det(A)=|A|= (2)(5)(1)=10
Note that if any of the elements of the main diagonal is zero then determinant would also be zero.

Process of Solving Inequalities

In mathematics both equalities and inequalities are present. The equalities give rise to equations. In equations the ‘equal to’ sign is used. There are various methods to solve the equations. The best and the easiest method must be used in order to arrive at the answer quickly and efficiently.

The concept of solving inequalities by addition and subtraction is very easy to understand and use. The concept can be very easy to understand if one is thorough with the concept of basic arithmetic operations like addition and subtraction.

The concept of inequality can be easily understood with the help of examples. The examples can help in explaining the concept very clearly.

In solving inequalities using addition or subtraction, the basic arithmetic operations of addition and subtraction are used. The inequality is called an inequality because the terms on the left side will not be equal to the terms on the right hand side. This is the basic reason for the term being called an inequality. This is the basic difference between the equality and inequality. In equality the terms on the left side will be equal to the terms on the right hand side.

The process of transposing will be used to solve the equations. Even in the case of inequalities the same process or principle is used. This is very simple principle. The terms on one side of the equation are moved to the others side; while doing so the sign of the term changes.

The positive term becomes the negative term and the negative term becomes the positive term. The multiplication becomes division and division becomes multiplication. So, one must be very careful while transposing the terms.

An inequality is formed when a term may be less on side or the other side. This is nothing but a side is surplus and the other side contains slack variable. So, the surplus has to be reduced to convert the inequality into equality.

But the same method can be used to solve both types of problems. Once the solutions are obtained for these types of problems it is necessary to check the feasibility of the solutions. All the solutions obtained might not satisfy the given equations. So, one has to be careful in selecting the final solutions. These solutions which are selected must satisfy the given equations. In case of inequality the solution must satisfy the inequality.

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