Matrices Algebra 1

Introduction to matrices algebra 1.

An  rectangular arrangement of numbers in rows and columns within the paranthesis is called a matrix. If there are m rows and n columns in a matrix then m `xx` n is called order of the matrix. In this topic let us know three operations on the given matrices.

Two matrices A and B can be added or subtracted if their orders are same. If not matrix addition or subtraction is not defined.

Two matrices can be multiplied if the number of column in the first matrix is equal to the number of rows in the second matrix. Otherwise the matrix multiplication is not defined.

Now let us see few problems in this topic matrices algebra 1

Examples of Matrices Algebra 1

Ex 1: Find the sum of the given matrices A and B.

A = `[[1,2],[3,4]]`    and B = `[[5,7],[6,8]]`

Soln: Since A and B are having same orders 2 `xx` 2,

A + B =  `[[1,2],[3,4]]` + `[[5,7],[6,8]]`

= `[[1+5,2+7],[3+6,4+8]]`   [Add the corresponding values]

Therefore A + B =  `[[6,9],[9,12]]`

Ex 2: Find the value of the given equation 2A – B.

Here A = `[[4,6],[7,5]]` and B = `[[3,2],[1,4]]`

Soln: 2A – B  = 2 `[[4,6],[7,5]]` - `[[3,2],[1,4]]`

= `[[2xx4, 2xx6],[2xx7,2xx5]]` - `[[3,2],[1,4]]`

= `[[8,12],[14,10]]` `[[3,2],[1,4]]`

= `[[8**3, 12**2],[14**1,10**4]]` [Subtract the corresponding values]

= `[[5,10],[13,6]]`

Therefore 2A – B =` [[5,10],[13,6]]`


More Example Problems on Matrices Algebra 1

Ex 3: Multiply the following matrices if possible:

(i)  A = `[[2,3,4],[5,6,7]]` and B  = `[[2,7],[5,6]]`

(ii) A = `[[3,4],[2,8]] `  and  B = `[[7,2],[1,8]]`

Soln: (i) The order of matrix

A = `[[2,3,4],[5,6,7]]` is 2 `xx` 3and B  = `[[2,7],[5,6]]`

For B, it is 2 `xx` 2.

Here the number of column in A is not equal to the number of rows in B (3≠ 2). Therefore AB is not possible.

(ii) Here in A and B, the number columns in A is equal number of rows in B (2=2). Therefore AB is possible

Therefore AB =  `[[3,4],[2,8]] ` `[[7,2],[1,8]]`

= `[[3 xx 7 + 4 xx 1,3 xx 2 + 4 xx 8],[2 xx 7 + 8 xx 1,2 xx 2 + 8 xx 8]]`

= `[[21 + 4,6 + 32],[14 + 8,4 + 64]]`

= `[[25,38],[22,68]]`

All Kinds of Math

Introduction to math:

Math is considered as the study of all quantity and all structure. Maths is considered as the tool in all fields. The math is mostly applicable in for the engineering and also for numerical analysis. The major kinds of the mathematics are algebra, trigonometry, statistics, probability and geometry. Here we are going to discuss about some kinds of math.

Kinds of Math

Kinds of math:

Algebra:

Algebra is considered as a kind of math which defines about the study of  all rules and also the operations. Algebra is also concerned with the equations, polynomials and all structure of algebra. This is considered as the main branch of mathematics. The algebra is classified into the various categories.

Statistics:

Statistics is a kind of math which makes use of a numerical data which is related to the group of individuals and experiments. Statistics deals with the collection, analysis and also the plan for the collection and analysis of data. Statistics includes the basic terms such as mean, median, Range, and mode.

Mean – process of finding the average of the numbers in the list.
Range – difference between the largest value and the smallest value in the list.
Median –finding the middle term from the list.
Mode – process of finding the frequently occurred term.
Trigonometry:

Trigonometry is the study of triangles which in particular defines the right triangles. This trigonometry takes place in both applied maths and also pure math which deals with the relations between the angle of the triangle and sides of the triangle. This includes three main functions such as sin, cos, tan.

Probability:

It is also a kind of math which defines the way for expressing the knowledge of events to occur. This probability is used in the areas such as statistics and finance.

Geometry:

Geometry is also a kind of math which defines the questions about the different kinds of shapes and also the study about the shapes. It is also related with the real life. Normal shapes used in geometry are triangle, circle, polygons, square and quadrilateral.

Example Problems

Example problem in algebra:

Solve: 2 (x+3) – (x+5) = 3(x+2) – (2x +5)

Solution:

2 (x+3) – (x+5) = 3(x+2) – (2x +5)

2x + (2*3) – (x+5) = 3x + (3 * 2) – 4x +7

2x + 6 – x + 5 = 3x + 6 -4x - 7

2x – x + 1 = 3x – 4x - 1

x + 1 = - x - 1

0 = 0

Hence the given equation is solved.

Example problems in statistics:

Find the mean, median, mode and range for the numbers 8, 6, 7, 4, 8, 5, 6, 8, 2.

Solution:

Mean:

Mean = `54/9` = 6.

Median:

Arrange the numbers in numerical order 2, 4, 5, 6, 6, 7, 8, 8, 8.

The term which is in middle is considered as the median. Therefore the answer is 6.

Range:

Range = maximum value – minmum value

Maximum value = 8.

Lowest value = 2.

Range = 8 – 2.

= 6.

Example in probability:

Three coins are tossed randomly. Find the sample space of getting 2 tails at the same time.

Solution:

Sample space = { HHH, HHT, HTT, HTH, THH, TTH, THT, TTT}

Probability of getting 2 tails = {TTH, THT, HTT, TTT}

= (Number of possible outcomes / Total number of outcomes}

= `4/8`

= `1/2`

Partial Products Math

Introduction of partial products math:

The partial products math is nothing but the numerical values and their operations in math. Partial products math topic involves the product of the natural numbers and the integers. Thus the partial products math not only deals with the products but also with the arithmetic operations like addition, subtraction and division. The whole numbers includes the natural numbers. Let capital N denotes the natural numbers.

Partial Products Math:

The partial products math is nothing but the summing of the two terms not only through the addition but also through the product method. First the terms are rounded and made to multiply with the left most term of the number. Then the second term is made to rounded and the values are made to multiplied with the another term of the left. Then the left most term can be made multiplied with left most term of another term. Then the right most term can be made to multiplied with the right most term of the another term.

At last the values are made to summing up and the total of the values gives the product of the both terms. This is how the product partial terms are executed. The partial math includes all the operations like addition and subtraction in the same manner. Then the second term is made to rounded and the values are made to multiplied with the another term of the left. Then the left most term can be made multiplied with left most term of another term.

Examples for the Partial Products Math:

Example1: Find the partial product of the term 83 x 27?



83
27
----
80*20 -> 1600
80* 7 ->  560
3*20 ->   60
3* 7 ->   21
----
2241
Answer: 2241

Example2: Find the partial product of the term 93 x 25?



93

25

--------

90*20 -> 1800

90*  5 ->   450

3*20 ->     60

3*  5 ->     15

---------

2325

Answer: 2325

Different Types of Math

Introduction mathematics:

The mathematic is deals with the logical calculation and quantitative calculation. The shape of the object is the arrangement, order, it has involved from the counting, measuring. The most significant branches of mathematics are algebra and analysis. A theoretical representative method used in the study of numbers, shapes, structure and change and the relationships. The different types of the math are the algebra, geometry, trigonometry, calculus, linear algebra, differential equations.

Different Type of the Math:

Algebra:

An algebra type is the branch of mathematics which treats of the associations and properties of measure by way of letters and other symbols. It is suitable to those associations that are correct of every class of magnitude.

Geometry:

Geometry types are the division of mathematics which investigate the relationships, property, and quantity of solids, surfaces, lines, and angles; the science which treat of the property and associations of magnitudes; the art of the relatives of space.

Trigonometry

A trigonometry types is the subdivision of the mathematics which is identified by the relative of the edges, and angle of the triangles. These kinds methods is give the assured position of the required position and also give the general relationship among the trigonometrically functions of arcs or angles.

Calculus:

The rate of probability is the calculus. The calculus can be divided into two type’s calculus; these are the differential calculus, and integral calculus. Differential calculus determines the rate of modify of a measure. Integral calculus is calculating the measuring rate of the change.

Additional Type of the Math:

Linear algebra:

The system of the solving is called as the linear equation. The linear conversion and vector spaces is the more frequent to the linear algebra. The equations are modifying as the function is the linear conversion. The system of the equation is the system of the transformation.

Differential equations:

A differential equation is a mathematical equation for an unidentified function of single or multiple variables that relate the ideals of the function itself and its derivatives of a variety of commands. The rate of the modification is used for the differential equation. They are most common a type of the differential equation is the ordinary differential equation and the partial differential equation.

Guided Reading Answers

Introduction to guided reading answers

The guided reading answers are nothing but getting help from others to reading the subjects.In online only we can get the help of tutor for reading any subjects.Initially the math problems can be solved by using some arithmetic operations  like addition,subtraction,division and multiplications and these can be denoted by (+ ,`xx` ,`-` ,÷ ).The following aticle shows some guided reading answers.

Solved Math Problems with some Guided Reading Answers

Problem 1:

Solve the 56x + 47y = 2632 given equation on the x and y intercepts.

Given:

56x + 47y = 2632

Solution:

56x + 47y = 2632

To find the x intercept of y = 0 and solve for x.

56x + 47(0) = 2632

Solve the value of x.

x = `2632/56`

x = 47

To find the y intercept of x=0 and solve for y.

56(0) + 47y = 2632

Solve the value of y

47y = 2632

y = `2632/47`

y = 56.

The equations of x intercept on (47,0) and y intercept on (0,56).

Problem 2:

Solve the problem 185 + 35 ( 40 + 27 ) ÷ 67 – 60 in method of order of operation
Solution:

Given:

`=>`  185 + 35 ( 40 + 27 ) ÷ 67 – 60

Step 1: we need to simplify the parentheses

`=>` 185 + 35 `xx` 67 ÷ 67 – 60

Step 2: We need to simplify the multiplication

`=>` 185 + 2345 ÷ 67 – 60

Step 3: We need to simplify the division

`=>` 185 + 35 – 60

Step 4: We need to simplify the addition

`=>`220 –  60

Step 5: We need to simplify the subtraction

`=>` 160

Answer: 185 + 35 ( 40 + 27 ) ÷ 67 – 60 = 160

Solved more Math Problems with some Guided Reading Answers

Problem 3:

Solve the given problem 11(s – 9) – 6s ` - ` 26 = 13(s + 33)
The Solutions follows below:

Step 1: Given expression is,

11(s – 9) – 6s ` - ` 26 = 13(s + 33)

Step 2:Multiplying the integer terms

11s – 99 – 6s – 26 = 13s + 429.

Step 3:Grouping the above terms

5s –125 = 13s + 429

Step 4: Add 125 on both sides

5s –125 + 125 = 13s + 429 + 125

Step 5:Grouping the above terms

5s = 13s + 554

Step 6:Subtract 13s by on both sides

5s `-` 13s = 13s `-` 13s + 554

Step 7:Grouping the above terms

–8s = 554

S = `- 554/8`

The required answers is

S = `- 554/8`

Problem 4:

Solve the given problem (156x2 – 141x – 91) + (213x2 – 181x – 144) `-` (–916x2 +   41x + 20)
Solution:

The problem can be solved in simplifying method .

Step 1:(156x2 – 141x – 91) + (213x2 – 181x – 144) `-` (–916x2 +   41x + 20)

Step 2: 156x2 – 141x – 91 + 213x2 – 181x – 144 + 916x2 `-`    41x `-` 20

Step 3:  1285x2 – 363x – 255

The required answer is

(156x2 – 141x – 91) + (213x2 – 181x – 144) `-` (–916x2 +   41x + 20) = 1285x2 – 363x – 255

Trigonometry Test Answers

Introduction to trigonometry test answers:

Trigonometry is an important branch of Mathematics. The word Trigonometry has been consequent from three Greek words Tri (Three), Goni (Angles), Metron (Measurement). Literally it means “measurement of triangles”. The fundamental trigonometric functions are Sine, Cosine and Tangent functions of a triangle takes an angle and give the sides of the triangle. The ordinary trigonometric terms are Sine, Cosine and Tangent.

Solving Trigonometric Functions on Trigonometry Test Answers:

Trigonometry test answers 1:

Solve the trigonometric equation.

(Find all solutions) 2 Cos x + 2 = 3

Solution:

First we have to solve for Cos x

2Cos x + 2 = 3

2Cos x =3 – 2

2Cos x = 1

Cos x = 1 / 2

X = Cos-1x (1/2)

X = 60

Trigonometry test answers 2:

Find the solutions for the trigonometric function:

-5 Cos 2x + 9 Sin x = -3

Solution:

-5 Cos2x + 9 Sin x = -3

-5(1- Sin2x) + 9Sinx = -3

-5 -5n2x + 9Sinx = -3

-5 -5Sin2x + 9Sinx +3 = 0

-5Sin2x + 9Sinx -2 = 0

Let us take y = Sin x

-5y2+ 9y – 2 = 0

Y = -2                                                     Y =-.2

Now plug y =sin x

Sin x = -2                                              Sin x = .2

X = Sin-1(-2)                                             X = Sin2(- .2)

X = - Sin-1(2)                                            X =-Sin-1(.2)

Trigonometry Test Answers 3 on Trigonometry Test Answers:

Prove (tanx+secx)/(cosx-tanx-secx)=-cscx

(tanx+secx) / (cosx-tanx-secx)
[(sinx+1)/cosx] / [(cos²(x)/cosx - sinx/cosx - 1/cosx]
[(sinx+1)/cosx] / [-(1-cos²x)/cosx - sinx/cosx]
[(sinx+1)/cosx] / [-sin²(x)/cosx - sinx/cosx]
[(sinx+1)/cosx] / [(-sin²(x)-sinx)/cosx]
[(sinx+1)/cosx]*[cosx/(-sin²(x)-sinx)]
(sinx+1)/-sin²(x)-sinx
sinx+1 / -sinx(sinx+1)
1/-sinx = -cscx

Trigonometry test answers 4:

Problem for trigonometric cosine function:

In a triangle adjacent side is 5 and hypotenuse is 25 then finds the angle A of the triangle?

Cos A= (adjacent) / (hypotenuse)

Cos A = 5/25

A = cos-1(5/25)

A = 78.46.

Trigonometry test answers 5:

Problem for trigonometric cosine function:

In a triangle adjacent side is 7 and hypotenuse is 2 then finds the angle A of the triangle?

Cos A= (adjacent) / (hypotenuse)

Cos A = 7/2

A = cos-1(7/2)

A = 0.998

How to Learn Fractions Exercises

In this article we shall discuss about how to learn fractions exercises. Here, fractions are also denoted as division of a whole. A fraction is can be creation over to a decimal all through dividing the upper digit, or numerator, during the lower digit, or denominator. Fractions are as an alternative of as ratios, and significance for fraction which is one of the main math processes. Thus the fractions `5/7` are also used to point out the ratio 5:7 and the fractions 5 ÷ 7 as well.



Example Problems Based on How to Learn Fractions Exercises:

The example problems based on how to learn fractions exercises are given below that,

Example 1:

How to learn fractions exercises of 129 divide by 5?

Solution:

Step 1:

Here, 129 divide by 5 is meant by `129/5` .

Step 2:

Now, 129 divide by 5 is given below that,

Here, using long division method. So, long division method is shown given below that,

25.8

5)129             [129 > 5, now divide `129/5` ]

125             [(hint: 25 × 5 = 125)]

40           [4 < 5, so add one zero]

40           [(hint: 8 × 5 = 40)]

0

Step 3:

The final answer is 25.8

Example 2:

How to learn fractions exercises of 467 divide by 13?

Solution:

Step 1:

Here, 467 divide by 13 is meant by `467/13.`

Step 2:

Now, 467 divide by 13 is given below that,

Here, using long division method. So, long division method is shown given below that,

35.92

13)467            [467 > 13, now divide `467/13`]

455            [(hint: 35 × 13 = 455)]

120          [12 < 13, so add one zero]

117          [(hint: 9 × 13 = 117)]

30        [3 < 13, so add one zero]

24        [(hint: 2 × 13 = 26)]

6

Step 3:

The final answer is 35.92





Practice Problems Based on How to Learn Fractions Exercises:

The practice problems based on how to learn fractions exercises are given below that,

Problem 1:

How to learn fractions exercises of 383 divide by 5?

Answer: The final answer is 76.6

Problem 2:

How to learn fractions exercises of 461 divide by 8?

Answer: The final answer is 57.625

Problem 3:

How to learn fractions exercises of 567 divide by 9?

Answer: The final answer is 63