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Jun 10, 2018

Worksheet on Addition of Matrices | Find the Sum of Two Matrices | Ans

Practice the problems given in the worksheet on addition of matrices. If M and N are the two matrices of the same order, then the matrices are said conformable for addition, and their sum is obtained by adding the corresponding elements of M and N. 1. Find the sum of A and B

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Jun 09, 2018

Properties of Scalar Multiplication of a Matrix |Scalar Multiplication

We will discuss about the properties of scalar multiplication of a matrix. If X and Y are two m × n matrices (matrices of the same order) and k, c and 1 are the numbers (scalars). Then the following results are obvious. I. k(A + B) = kA + kB II. (k + c)A = kA + cA III. k(cA)

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Jun 07, 2018

Scalar Multiplication of a Matrix | Examples on Scalar Multiplication

The operation of multiplying variables by a constant scalar factor may properly be called scalar multiplication and the rule of multiplication of matrix by a scalar is that the product of an m × n matrix A = [aij] by a scalar quantity c is the m × n matrix [bij] where bij

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Jun 06, 2018

Subtraction of Matrices | Examples on Difference of Two Matrices

We proceed to develop the algebra of subtraction of matrices. Two matrices A and B are said to be conformable for subtraction if they have the same order (i.e. same number of rows and columns) and their difference A - B is defined to be the addition of A and (-B).

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Jun 05, 2018

Properties of Addition of Matrices | Commutative Law | Associative Law

We will discuss about the properties of addition of matrices. 1. Commutative law of addition of matrix: Matrix multiplication is commutative. This says that, if A and B are matrices of the same order such that A + B is defined then A + B = B + A. Proof: Let A = [aij]m × n

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Jun 04, 2018

Addition of Matrices | Example on Sum of Two Matrices

We proceed to develop the algebra of matrices. Two matrices A and B are said to be conformable for addition if they have the same order (same number of rows and columns). If A = (aij)m, n and B = (bij)m,n then their sum A + B is the matrix C = (cij)m,n where cij = aij + bij

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Jun 01, 2018

Triangular Matrix | Upper Triangular Matrix | Lower Triangular Matrix

There are two types of triangular matrices. 1. Upper Triangular Matrix: A square matrix (aij) is said to be an upper triangular matrix if all the elements below the principal diagonal are zero (0). That is, [aij]m × n is an upper triangular matrix if (i) m = n and (ii) aij

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May 29, 2018

Height and Distance with Two Angles of Elevation | Solved Problems

We will solve different types of problems on height and distance with two angles of elevation. Another type of case arises for two angles of elevations. In the given figure, let PQ be the height of pole of ‘y’ units. QR be the one of the distance between the foot of the pole

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May 22, 2018

Angle of Elevation | How to Find out the Angle of Elevation

We have already learnt about trigonometry in previous units in detail. Trigonometry has its own applications in mathematics and in physics. One such application of trigonometry in mathematics is “height and distances”. To know about height and distances, we have to start

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May 20, 2018

Identity Matrix | Unit Matrix |If [d] is a scalar matrix then [d] = dI

A scalar matrix whose diagonal elements are all equal to 1, the identity element of the ground field F, is said to be an identity (or unit) matrix. The identity matrix of order n is denoted by In. A scalar matrix is said to be a unit matrix, if diagonal elements are unity.

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May 17, 2018

Definition of Equal Matrices | Examples of Equal Matrices

Equality of two matrix: Two matrices [aij] and [bij] are said to be equal when they have the same number of rows and columns and aij = bij for all admissible values of i and j. Definition of Equal Matrices: Two matrices A and B are said to be equal if A and B have the same

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May 10, 2018

Null Matrix | Null or Zero Matrix|Zero Matrix|Problems on Null Matrix

If each element of an m × n matrix be 0, the null element of F, the matrix is said to be the null matrix or the zero matrix of order m × n and it is denoted by Om,n. It is also denoted by O, when no confusion regarding its order arises. Null or zero Matrix: Whether A is a

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May 07, 2018

Column Matrix | Definition of Column Matrix |Examples of Column Matrix

Here we will discuss about the column matrix with examples. In an m × n matrix, if n = 1, the matrix is said to be a column matrix. Definition of Column Matrix: If a matrix have only one column then it is called column matrix. Examples of column matrix:

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May 03, 2018

Row Matrix | Definition of Row Matrix | Examples of Row Matrix

In an m × n matrix, if m = 1, the matrix is said to be a row matrix. Definition of Row Matrix: If a matrix have only one row then it is called row matrix. Here we will discuss about the row matrix with examples. Examples of row matrix:

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Apr 25, 2018

Square Matrix | Definition of Square Matrix |Diagonal of Square Matrix

If square matrixes have n rows or columns then the matrix is called the square matrix of order n or an n-square matrix. Definition of Square Matrix: An n × n matrix is said to be a square matrix of order n. In other words when the number of rows and the number of columns in

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Apr 25, 2018

Matrix | Definition of a Matrix | Examples of a Matrix | Elements

A rectangular array of mn elements aij into m rows and n columns, where the elements aij belongs to field F, is said to be a matrix of order m × n (or an m × n matrix) over the field F. Definition of a Matrix: A matrix is a rectangular arrangement or array of numbers

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Mar 30, 2018

Joint Variation | Solving Joint Variation Problems and Application

One variable quantity is said to vary jointly as a number of other variable quantities, when it varies directly as their product. If the variable A varies directly as the product of the variables B, C and D, i.e., if.A ∝ BCD or A = kBCD (k = constant ), then A varies jointly

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Mar 30, 2018

Indirect Variation | Inverse Variation | Inverse or Indirect Variation

When two variables change in inverse proportion it is called as indirect variation. In indirect variation one variable is constant times inverse of other. If one variable increases other will decrease, if one decrease other will also increase. This means that the variables

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Mar 26, 2018

Direct Variation | Solving Direct Variation Word Problem

When two variables change in proportion it is called as direct variation. In direct variation one variable is constant times of other. If one variable increases other will increase, if one decrease other will also decease. This means that the variables change in a same ratio

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Mar 25, 2018

Methods of Solving Simultaneous Linear Equations | Solved Examples

There are different methods for solving simultaneous linear Equations: 1. Elimination of a variable 2. Substitution 3. Cross-multiplication 4. Evaluation of proportional value of variables This topic is purely based upon numerical examples. So, let us solve some examples

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Mar 24, 2018

Method of Cross Multiplication|Solve by Method of Cross Multiplication

The next method of solving linear equations in two variables that we are going to learn about is method of cross multiplication. Let us see the steps followed while soling the linear equation by method of cross multiplication: Assume two linear equation be A1 x + B1y + C1=

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Mar 06, 2018

Properties of Angles of a Triangle |Sum of Three Angles of a Triangle

We will discuss about some of the properties of angles of a triangle. 1. The three angles of a triangle are together equal to two right angles. ABC is a triangle. Then ∠ZXY + ∠XYZ + ∠YZX = 180° Using this property, let us solve some of the examples. Solved examples

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Mar 02, 2018

Geometrical Property of Altitudes|Altitudes of Triangle are Concurrent

The three altitudes of triangle are concurrent. The point at which they intersect is known as the orthocentre of the triangle. In the adjoining figure, the three altitudes XP, YQ and ZR intersect at the orthocentre O.

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Mar 01, 2018

Medians and Altitudes of a Triangle |Three Altitudes and Three Medians

Here we will discuss about Medians and Altitudes of a Triangle. Median: The straight line joining a vertex of a triangle to the midpoint of the opposite side is called a median. A triangle has three medians. Here XL, YM and ZN are medians. A geometrical property of medians

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Feb 28, 2018

Classification of Triangles on the Basis of Their Sides and Angles

Here we will discuss about classification of triangles on the basis of their sides and angles Equilateral triangle: An equilateral triangle is a triangle whose all three sides are equal. Here, XYZ is an equilateral triangle as XY = YZ = ZX. Isosceles triangle: An isosceles

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Feb 22, 2018

Triangle | Exterior Opposite Angles|Interior Opposite Angles|Perimeter

A triangle is a plane figure bounded by three straight lines. A triangle has three sides and three angles, and each one of them is called an element of the triangle. Here, PQR is a triangle, its three sides are line segments PQ, QR and RP; ; ∠PQR, ∠QRP and ∠RPQ are its

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Feb 14, 2018

Problem on Change the Subject of a Formula | Changing the Subject

We will solve different types of problems on change the subject of a formula. The subject of a formula is a variable whose relation with other variables of the context is sought and the formula is written in such a way that subject is expressed in terms of the other

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Feb 13, 2018

Worksheet on Change of Subject | Change the Subject as Indicated

Practice the questions given in the worksheet on change of subject When a formula involving certain variables is known, we can change the subject of the formula. What is the subject in each of the following questions? Change the subject as indicated.

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Feb 02, 2018

Worksheet on Framing a Formula | Framing Formulas | Frame an Equation

Practice the questions given in the worksheet on framing a formula. I. Frame a formula for each of the following statements: 1. The side ‛s’ of a square is equal to the square root of its area A.

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Jan 31, 2018

Establishing an Equation | Framing a Formula | Framing Linear Equation

We will discuss here about establishing an equation. In a given context, the relation between variables expressed by equality (or inequality) is called a formula. When a formula is expressed by an equality, the algebraic expression is called an equation.

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Jan 22, 2018

Rectangular Cartesian Co-ordinates | Abscissa | Ordinate | Oblique Co-ordinate

What is Rectangular Cartesian Co-ordinates? Let O be a fixed point on the plane of this page; draw mutually perpendicular straight line XOX’ and YOY’ through O. Clearly, these lines divide the plane

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Jan 21, 2018

What is Co-ordinate Geometry? | Analytical Geometry| Cartesian Co-ordinate

What is co-ordinate geometry? The subject co-ordinate geometry is that particular branch of mathematics in which geometry is studied with the help of algebra. This branch of mathematics was fir

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Jan 20, 2018

Table of Tangents and Cotangents | Natural Tangents and Natural Cotangents

We will discuss here the method of using the table of tangents and cotangents. This table shown below is also known as the table of natural tangents and natural cotangents. Using the table we can

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Jan 20, 2018

Table of Sines and Cosines |Trigonometric Table|Table of Natural sines & cosines

We will discuss here the method of using the table of sines and cosines: The above table is also known as the table of natural sines and natural cosines. Using the table we can find the values

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Jan 20, 2018

Problems on Properties of Triangle | Angle Properties of Triangles

We will solve different types of problems on properties of triangle. 1. If in any triangle the angles be to one another as 1 : 2 : 3, prove that the corresponding sides are 1 : √3 : 2.

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Jan 20, 2018

Properties of Triangle Formulae | Triangle Formulae | Properties of Triangle

We will discuss the list of properties of triangle formulae which will help us to solve different types of problems on triangle.

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Jan 20, 2018

Law of Tangents |The Tangent Rule|Proof of the Law of Tangents|Alternative Proof

We will discuss here about the law of tangents or the tangent rule which is required for solving the problems on triangle. In any triangle ABC,

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Jan 20, 2018

Area of a Triangle | ∆ = ½ bc sin A | ∆ = ½ ca sin B | ∆ = ½ ab sin C

If ∆ be the area of a triangle ABC, Proved that, ∆ = ½ bc sin A = ½ ca sin B = ½ ab sin C That is, (i) ∆ = ½ bc sin A (ii) ∆ = ½ ca sin B (iii) ∆ = ½ ab sin C

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Jan 20, 2018

The Law of Cosines | The Cosine Rule | Cosine Rule Formula | Cosine Law Proof

We will discuss here about the law of cosines or the cosine rule which is required for solving the problems on triangle. In any triangle ABC, Prove that, (i) b\(^{2}\)

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Jan 20, 2018

Proof of Projection Formulae | Projection Formulae | Geometrical Interpretation

The geometrical interpretation of the proof of projection formulae is the length of any side of a triangle is equal to the algebraic sum of the projections of other sides upon it. In Any Triangle

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Jan 20, 2018

Projection Formulae | a = b cos C + c cos B | b = c cos A + a cos C

Projection formulae is the length of any side of a triangle is equal to the sum of the projections of other two sides on it. In Any Triangle ABC, (i) a = b cos C + c cos B

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Jan 20, 2018

Theorem on Properties of Triangle | p/sin P = q/sin Q = r/sin R = 2K

Proof the theorem on properties of triangle p/sin P = q/sin Q = r/sin R = 2K. Proof: Let O be the circum-centre and R the circum-radius of any triangle PQR. Let O be the circum-centre and R

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Jan 20, 2018

The Law of Sines | The Sine Rule | The Sine Rule Formula | Law of sines Proof

We will discuss here about the law of sines or the sine rule which is required for solving the problems on triangle. In any triangle the sides of a triangle are proportional to the sines

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Jan 20, 2018

Properties of Triangles | Semi-perimeter| Circum-circle|Circum-radius|In-radius

In trigonometry we will discuss about the different properties of triangles. We know any triangle has six parts, the three sides and the three angles are generally called the elements of the triangle.

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Jan 19, 2018

Problems on Inverse Trigonometric Function | Inverse Circular Function Problems

We will solve different types of problems on inverse trigonometric function. 1. Find the values of sin (cos\(^{-1}\) 3/5)

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Jan 19, 2018

Principal Values of Inverse Trigonometric Functions |Different types of Problems

We will learn how to find the principal values of inverse trigonometric functions in different types of problems. The principal value of sin\(^{-1}\) x for x > 0, is the length of the arc of a unit

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Jan 19, 2018

Inverse Trigonometric Function Formula | Inverse Circular Function Formula

We will discuss the list of inverse trigonometric function formula which will help us to solve different types of inverse circular or inverse trigonometric function.

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Jan 19, 2018

3 arctan(x) | 3 tan\(^{-1}\) x |3 tan inverse x | Inverse Trigonometric Function

We will learn how to prove the property of the inverse trigonometric function 3 arctan(x) = arctan(\(\frac{3x - x^{3}}{1 - 3 x^{2}}\)) or, 3 tan\(^{-1}\) x = tan\(^{-1}\)

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Jan 19, 2018

3 arccos(x) | 3 cos\(^{-1}\) x |3 cos inverse x | Inverse Trigonometric Function

We will learn how to prove the property of the inverse trigonometric function 3 arccos(x) = arccos(4x\(^{3}\) - 3x) or, 3 cos\(^{-1}\) x = cos\(^{-1}\) (4x\(^{3}\) - 3x)

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