Simplest and shortest Matrix method to solve linear equations of 3 variables

Matrix method to solve linear equations of 3 variables

In this post we are going to understand the concept of solving linear equations of three variables with the help of matrix method .

Matrix method to solve linear equations of three variables with the help of example. 

Set of given equations are 

x - y + z = 2    

2x - y = 0  

 2y - 2 z = 1  

Rearranging these equations in symmetrical form. It means if any one of the variable in any equation is missing then write that missing variable/s with zero coefficient. As we see in this case the coefficient of z in 2nd equation and the coefficient of x in 3rd equation are missing. So we have to write coefficient of x in 2nd equation and coefficient of x in 3rd equation as zero.

x - y + z = 2 ----------------------> (1)

2x - y + 0z = 0 ------------------> (2)

0.x + 2y - 2z = 1 ----------------> (3)

The system of these equations can be transformed into Matrix form .

AX = B , ⇒ X = A^-1B  -------> (*)

Where A is matrix written from the coefficients of x, y and z when these equations are in symmetric form and B is the matrix written from constants from right hand sides in column form and X is matrix of all the variables in column form.

In order to find the solution of set of these equations , first we have to find the inverse of matrix A if it exist then we can find the solution otherwise Matrix method fails to find the solution of the set of linear equations . 

Evaluation of Determinant 

                                                  

|A| = 4 (18 - 4) -0(9 - 3) +6(12 - 18)

       = 4(14) + 0 + 6(-6) 

       = 56 - 36

        = 20

Since the determinant value of this matrix is not equal to zero ,Therefore its inverse can be calculated.

And formula for finding the inverse of matrix A is

Where Adjoint A is the transpose of co factor matrix. And in order to find the co factor matrix of any matrix, we have to find co factors of all the elements present in this matrix. 

How to calculate  co factors of all the elements of the matrix A

Let us calculate these cofactors.  

Now these co factors can be written in matrix form known as co factor Matrix. 

Co factors of 1st row are  (18 - 4) , -(9 - 3), (12 - 18) 

i. e. Co factors of 1st row are 14, -6 , -6

Co factors of 2nd row are -(0 -24), (12 - 18) , -(16 - 0) 

I. e. Co factors of 2nd row are  24 , -6 , -16

Co factors of 3rd row are  (0 - 36), -(4 - 18), (24 - 0) 

i.e. Co factors of 3rd row are  -36 , 14 , 24

Co factor Matrix

Writing co factors of 1st row in 1st row of this matrix , co factors of 2nd row in 2nd row of this matrix . Similarly co factors of 3rd row in 3rd row of this matrix . 

 

Adjoint  Matrix

To find the Ad joint of this matrix we have to take it's transpose, Because transpose of any matrix is called Ad joint of the matrix. So writing all the elements which are in 1st row in 1st column, and  all the elements which are in 2nd row in 2nd column and  all the elements which are in 3rd row in 3rd column. 

Inverse  Matrix

Now we can find inverse of the matrix A by putting the value of inverse of A in equation  (4), Now  putting the values  Matrix B and    A^-1  in (4) After simplification and using the properties of equality of two matrices  ( Two matrices of same order are equal iff their respective elements are equal to each other ) 

Now we shall use the property of equality of two matrices ,which says that if two matrices are equal to each other then their respective elements must be  equal to each other.

⇒ x = 10

    y = 10

    z = 10

So this was the Matrix method of solving linear equations of three variables using inverse of matrix. Your valuables comments will be appreciated for betterment of this blog.

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Inverse method of solving linear equations of three variables

Inverse method of solving linear equations of 3 variables

Matrix method of solving linear equations of three variables with the help of example. 

Set of given equations are 

4x + 6z = 100     --------------------------------> (1)

3x + 6y + z = 100   -----------------------------> (2)

3x + 4y + 3z  = 100   ---------------------------> (3)

The system of these equations can be transformed into Matrix formed 

AX =  B  ,  ⇒ X =   A^-1 B    -------------------------->  (*)

Where A is matrix written from the coefficients of x, y and z when these equations are in symmetric form and B is the matrix written from constants from right hand sides in column form and X is matrix of all the variables in column form. 

In order to find the solution of set of these equations , first we have to find the inverse of matrix A if it exist then we can find the solution otherwise Matrix method fails to find the solution of the set of linear equations . 

Evaluation of Determinant 

|A|  = 4 (18 - 4) -0(9 - 3) +6(12 - 18)

       =  4(14) + 0 + 6(-6) 

       = 56 - 36

        = 20

Since the determinant value of this matrix  is not equal to zero ,Therefore its inverse can be calculated.

And  formula for finding the inverse of matrix A is 

Where Adjoint A is the transpose of co factor matrix. And in order to find the co factor matrix of any matrix, we have to find co factors of all the elements present in this matrix. 

How to calculate  co factors of all the elements of the matrix A

Let us calculate these cofactors.  

Now these co factors can be written in matrix form known as co factor Matrix. 

Co factors of 1st row are  (18 - 4) , -(9 - 3), (12 - 18) 

i. e. Co factors of 1st row are 14, -6 , -6

Co factors of 2nd row are -(0 -24), (12 - 18) , -(16 - 0) 

I. e. Co factors of 2nd row are  24 , -6 , -16

Co factors of 3rd row are  (0 - 36), -(4 - 18), (24 - 0) 

i.e. Co factors of 3rd row are  -36 , 14 , 24

Co factor Matrix

Writing co factors of 1st row in 1st row of this matrix , co factors of 2nd row in 2nd row of this matrix . Similarly co factors of 3rd row in 3rd row of this matrix . 

Adjoint  Matrix

To find the Ad joint of this matrix we have to take it's transpose, Because transpose of any matrix is called Ad joint of the matrix. So writing all the elements which are in 1st row in 1st column, and  all the elements which are in 2nd row in 2nd column and  all the elements which are in 3rd row in 3rd column. 

Inverse  Matrix

Now we can find inverse of the matrix A by putting the value of inverse of A in equation  (4), we get

Now  putting the values  Matrix B and    A^-1  in (4) 

After simplification and using the properties of equality of two matrices  ( Two matrices of same order are equal iff their respective elements are equal to each other ) 

⇒  x = 10

     y  = 10

      z = 10

So this was the Matrix method of solving linear equations of three variables using inverse of matrix. Your valuables comments will be appreciated for betterment of this blog.

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Matrix method of solving linear equations of three variables

Learn the process of solving linear equations of three variables by matrix method  .Let us understand this method with the help of an example

Matrix method of solving linear equations of 3 variables

Matrix method of solving linear equations of three variables with the help of an example. 

The system of these equations can be transformed into Matrix form as 

AX =  B  ,  ⇒ X =   A^-1 B   ->  (*)

Where A is matrix written from the coefficients of x, y and z when these equations are in symmetric form and B is the matrix written from constants from right hand sides in column form and X is matrix of all the variables in column form. 

In order to find the solution of set of these equations , first we have to find the inverse of matrix A if it exist then we can find the solution otherwise Matrix method fails to find the solution of the set of linear equations . 

Evaluation of Determinant 

|A|  = 1 (-9 - 27) -1(6 - 63) -1(6 + 21)

       =  -36 + 57 - 27

       = -63 + 57

        = -6

Since the determinant value is not equal to zero ,Therefore its inverse can be calculated.

And  formula for finding the inverse of matrix A is 

Where Adjoint A is the transpose of co factor matrix. And in order to find the co factor matrix of any matrix, we have to find co factors of all the elements present in that matrix. 

How to calculate  co factors of all the elements of the matrix A. 

 Let us calculate these cofactors. 

Co factors of 1st row are  -36 , 57 , 27

Co factors of 2nd row are  -6, 10 , 4

Co factors of 3rd row are  6 . -11 , -5

Now these co factors can be written in matrix form known as co factor Matrix. 

Co factor Matrix

Writing co factors of 1st row in 1st row of this matrix , co factors of 2nd row in 2nd row of this matrix . Similarly co factors of 3rd row in 3rd row of this matrix . 

Adjoint  Matrix

To find the Ad joint of this matrix we have to take it's transpose, Because transpose of any matrix is called Ad joint of the matrix. So writing all the elements which are in 1st row in 1st column, and  all the elements which are in 2nd row in 2nd column and  all the elements which are in 3rd row in 3rd column. 

Now we can find inverse of the matrix A by putting the value of inverse of A in equation  (4), we get

Now  putting the values  Matrix B and    A^-1  in (4) 

After simplification and using the properties of equality of two matrices  ( Two matrices of same order are equal if and only if their respective elements are equal to each other ) 

⇒  x = -54/-6 = 9

 y =  12/-6 = 2

 z  = -24/-6  = -4

Hence

 x = 9 

 y = 2

 z = -4

So this was the Matrix method of solving linear equations of three variables using inverse of matrix. Your valuables comments will be appreciated for betterment of this blog.

Also read this post for understanding inverse of matrix using elementary row transformation

 

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Finding inverse of 3x3 matrix using elementary transformation

We shall learn the process of finding inverse of 3x3 matrix using elementary transformation . The  method of finding inverse of the matrix 3 x 3 using elementary row transformations involves 5 to 6 steps . Let us understand this method with the help of an example.

Inverse of 3x3 matrix using elementary transformation

For finding Inverse of  3 x 3 Matrix using Elementary Row Transformations , We shall strat with this formula

  A  = I A ,

 Where I is Unit Matrix of order 3

In the 1st step we have to make 1st element of 1st row and 1st column unity. We shall start changing the given matrix written  in left hand side  step by step to unit matrix and this this process will also appliesd to the the unit matrix written on right hand side of equation (1) , Hence this change the  matrix written on the right hand side of  equation (1) to other matrix . And when the given matrix on the left hand side changed to Unit matrix , the matrix which was on the right hand side of equation of (1) will be inverse of the given matrix.

Suggested direction of elementary rows operations  are as follows

In the 1st step we have to make the 1st element of 1st column to unity by using taking suitable number common from it .

In the next step by using the above step we have to make 2nd and 3rd  element of 1st column unity by using suitable row transformations. 

 

Till now we have made two elements of 1st column zero , now we have to make 1st and 2nd elements of 3rd Row zero by using suitable row elementary transformations. Note carefully 3rd elements of 3rd row can not be changed to zero as this element will have to  be reduced to unity as in Unit Matrix. 

After this step start making the elements of 2nd Row equal to zero but using suitable row transformations.

Now we shall conclude the process of inverse of matrix  with the changing of  2nd and 3rd elements of 1st row  to zero.. 

Here matrix B is the inverse of the given matrix A. 

To verify that the matrix B obtained is correct or not. We can check it by multiplying matrix B with I, if it comes out equal to matrix A. Then our answer is correct. 

Also read this Post for solution of set of three variables equations using matrix method

So the process of finding inverse of 3x3 matrix using elementary transformation discussed in this post with the help of an example. Your comment will always be appreciated for betterment of this blog. 

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How to find proportion of Four numbers

How to find proportion of 4 numbers, proportion of Four numbers with the help of examples

What is Proportion

Proportion is majorly based on ratio and fractions. A fraction is written in the form of a/b, while ratio a:b, then a proportion means that two ratios are equal. Here a and b are any two numbers. The ratio and proportion are key foundations to understand the various concepts in mathematics .

Proportion can be applied to solve many daily life problems based on mixture ,work and time , time and speed etc . It develops a relation between two or more quantities and thus helps in their comparison.

What is Proportion?

Proportion in general is referred to as a part, share, or number considered in comparative relation to a whole. Proportion says that when two ratios are equivalent, they are in proportion. It is an equation or statement used to depict that two ratios or fractions are equal .

How to find proportion of Four numbers

If A : B = 2 : 3, B : C = 4 : 5 and      C : D = 6 : 7,    what is A  : D  ?

How to find A : B : C : D

Let  us mark these columns 1st and 2nd  as follows 

                      1   2

A   :    B =     2    :   3 

B    :     C =    4    :   5

C     :     D =    6    : 7

The value of A can be calculated by taking product  of all numbers  appearing under column (1).Therefore  A is product of  2 , 4 and 6

A = 2 x 4 x 6 = 48

The value of B can be calculated by  product under column (1) beginning from second row and numbers  under column (2) for all previous rows. Therefore  B is product of  4 , 6 and 3

B = 4 x 6 x 3 = 72

Similarly C can be computed  by taking  product of all numbers  under column (1) beginning from third row and terms under column (2) for all previous rows.Hence  C is product of  6 ,3 and 5

C = 6 x 3 x 5 = 90

D is result  of all terms appearing under column (2).

D = 3 x 5 x 7 = 105

=> A:B:C:D = 48 : 72 : 90 : 105

Ratio between any two or three variables from above.

A  : D = 48  :   105 = 16  :  35
This logic can be applied  to any number of rows

If A:B,  B:C,  C:D, D:E and E:F are  given, we can find A : B : C : D : E : F using above method.

Example :

A:B = 1:2

B:C = 2:7

C:D = 3:5

D:E = 3:8

E:F = 1:3

To find value of  A

To find out value of A multiply all the numbers which are under 1st column starting from this Row . We must choose one number less  to multiply as the variables ( A, B, C ,D and E ).

A = 1 × 2 × 3 × 3 ×1 = 18

To find value of  B

To find out value of B multiply all the numbers which are under 1st column starting from this Row  ( here four numbers are in 1st column which are below  A ,so our 5th number will be 1st number from column 2nd.

B = 2 × 3 × 3 × 1 × 2 = 36

To find value of  C

To find out value of C multiply  all the numbers which are under 1st column starting from this Row ( here three numbers are in 1st column which are below  B ,so our 4th and 5th numbers  will be 1st and 2nd numbers  from column 2nd.

C =  3 × 3 × 1 × 2 × 7 =126

To find value of  D

To find out value of C multiply  all the numbers which are under 1st column starting from this Row ( here two numbers are in 1st column which are below  C , so our 3rd
,  4th  and 5th numbers will be 1st , 2nd and 3rd numbers  from column 2nd .

D = 3 × 1 × 2 × 7 × 5 = 210

To find value of  E

To find out value of C multiply  all the numbers which are under 1st column starting from this Row ( here 1st  numbers is in 1st column which are below  D ,so our 2nd ,3rd , 4th and 5th number will be 1st ,2nd ,3rd and 4th number from column 2nd.

E = 1 × 2 × 7 × 5 × 8 = 560

To find value of  F

To find out value of F multiply all the numbers which are under 2nd column starting in front of  1st Row . We must choose one number less  to multiply as the variables ( A, B, C ,D and E ).

F = 2 × 7 × 5 × 8 × 3 = 1680

     A  :   B   :   C   :      D  :     E   :  F

= 18  :  36  : 126 : 210   : 560   : 1680
Now dividing  by 2 every terms,  we have 

= 9   : 18  : 63  : 105   :  280   :  840

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How to find shortest distance between two lines

SHORTEST DISTANCE BETWEEN TWO LINES

1st of all we shall find out shortest distance between two Parallel lines.

Problem 1

Consider two parallel lines whose equations in vector form are given by

 Now comparing these equations with standard form , and write

 _, and  vectors ,we get

Now applying this formula to find the shortest distance between two lines .

As it is clear from formula , we have to find cross product of  and  and then magnitude of vector 

Now find the magnitude of  × vector

                                 =√(81)+(196)+(16)

                                 =√293

Magnitude of 

                    = √49

           = 7

Putting all these values in  the formula of Shortest Distance between two lines .

Now Find distance between two skew lines i.e. Lines which are not Parallel lines.

Problem 2

Consider two parallel lines whose equations in vector form are given by

 Now comparing these equations with standard form , and write

 _, and  vectors ,we get

 Now applying this formula to find the shortest distance between two lines 

 Now find cross product and then  magnitude of these two vectors

Don't Forget Watch this Video to understand better

FINAL WORDS

Thanks for watching and responding to this post "HOW TO FIND SHORTEST DISTANCE BETWEEN TWO LINES".

My previous Posts

HOW TO SOLVE LINEAR EQUATIONS OF THREE  VARIABLES  BY  MATRIX  METHOD

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How to find common area of two parabolas.

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