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Matrices are distinguished on the basis of their order, elements and certain other conditions. There are different types of matrices but the most commonly used are discussed below. Let’s find out the types of matrices in the field of mathematics.
Types of Matrices
Different types of Matrices and their forms are used for solving numerous problems. Some of them are as follows:
1) Row Matrix
A row matrix has only one row but any number of columns. A matrix is said to be a row matrix if it has only one row. For example,
A=[−1/2√523]
is a row matrix of order 1 × 4. In general, A = [aij]1 × n is a row matrix of order 1 × n.2) Column Matrix
A column matrix has only one column but any number of rows. A matrix is said to be a column matrix if it has only one column. For example,
A=⎡⎣⎢⎢⎢0√3−11/2⎤⎦⎥⎥⎥
is a column matrix of order 4 × 1. In general, B = [bij]m × 1 is a column matrix of order m × 1.3) Square Matrix
Different types of Matrices and their forms are used for solving numerous problems. Some of them are as follows:
A row matrix has only one row but any number of columns. A matrix is said to be a row matrix if it has only one row. For example,
is a row matrix of order 1 × 4. In general, A = [aij]1 × n is a row matrix of order 1 × n.
A column matrix has only one column but any number of rows. A matrix is said to be a column matrix if it has only one column. For example,
is a column matrix of order 4 × 1. In general, B = [bij]m × 1 is a column matrix of order m × 1.
A square matrix has the number of columns equal to the number of rows. A matrix in which the number of rows is equal to the number of columns is said to be a square matrix. Thus an m × n matrix is said to be a square matrix if m = n and is known as a square matrix of order ‘n’. For example,
is a square matrix of order 3. In general, A = [aij] m × m is a square matrix of order m.
A square matrix has the number of columns equal to the number of rows. A matrix in which the number of rows is equal to the number of columns is said to be a square matrix. Thus an m × n matrix is said to be a square matrix if m = n and is known as a square matrix of order ‘n’. For example,
A=⎡⎣⎢33/24−1√3/2301−1⎤⎦⎥
is a square matrix of order 3. In general, A = [aij] m × m is a square matrix of order m.
In Mathematics, Differentiation can be defined as a derivative of a function with respect to an independent variable. Differentiation, in calculus, can be applied to measure the function per unit change in the independent variable.
Let y = f(x) be a function of x. Then, the rate of change of “y” per unit change in “x” is given by:
dy / dx
If the function f(x) undergoes an infinitesimal change of ‘h’ near to any point ‘x’, then the derivative of the function is defined as
If we are given with real valued function (f) and x is a point in its domain of definition, then the derivative of function, f, is given by:
f'(a) = limh→0[f(x + h) – f(x)]/h
provided this limit exists.
Let us see an example here for better understanding.
Example: Find the derivative of f(x) = 2x, at x =3.
Solution: By using the above formulas, we can find,
f'(3) = limh→0 [f(3 + h) – f(3)]/h = limh→0[2(3 + h) – 2(3)]/h
f'(3) = limh→0 [6 + 2h – 6]/h
f'(3) = limh→0 2h/h
f'(3) = limh→0 2 = 2Notations
When a function is denoted as y = f(x), the derivative is indicated by the following notations.
D(y) or D[f(x)] is called Euler’s notation.
dy/dx is called Leibniz’s notation.
F’(x) is called Lagrange’s notation.
The meaning of differentiation is the process of determining the derivative of a function at any point.
Linear and Non-Linear Functions
Functions are generally classified into two categories under Calculus, namely:
(i) Linear functions
(ii) Non-linear functions
A linear function varies at a constant rate through its domain. Therefore, the overall rate of change of the function is the same as the rate of change of a function at any point.
However, the rate of change of function varies from point to point in the case of non-linear functions. The nature of variation is based on the nature of the function.
The rate of change of a function at a particular point is defined as a derivative of that particular function.
Differentiation Formulas
The important Differentiation formulas are given below in the table. Here, let us consider f(x) as a function and f'(x) is the derivative of the function.
If f(x) = tan (x), then f'(x) = sec2x
If f(x) = cos (x), then f'(x) = -sin x
If f(x) = sin (x), then f'(x) = cos x
If f(x) = ln(x), then f'(x) = 1/x
If f(x) = ex, then f'(x) = ex
If f(x) = xn, where n is any fraction or integer, then f'(x) = nxn-1
If f(x) = k, where k is a constant, then f'(x) = 0
trigonometry, the branch of mathematics concerned with specific functions of angles and their application to calculations. There are six functions of an angle commonly used in trigonometry. Their names and abbreviations are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). These six trigonometric functions in relation to a right triangle are displayed in the figure. For example, the triangle contains an angle A, and the ratio of the side opposite to A and the side opposite to the right angle (the hypotenuse) is called the sine of A, or sin A; the other trigonometry functions are defined similarly. These functions are properties of the angle A independent of the size of the triangle, and calculated values were tabulated for many angles before computers made trigonometry tables obsolete. Trigonometric functions are used in obtaining unknown angles and distances from known or measured angles in geometric figures.
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