how can we calculte inverse matrix?

Question

related to mathematics

Answer 1

it is very simple
first of all take a matrix
eg. [1 1 2]
A = [1 2 1]
[2 1 1]
now first of all find the determinant of A i.e. |A| which is
[ 1 * { (2 * 1) - (1 * 1) } ] - [ 1 * { (1 * 1) - (2 * 1) } ] + [2 * { (1 * 1) - (2 * 2) } ] which is equal to - 4 is it?
now
find cofactor matrix M,
m1 = (2 * 1) - (1 * 1) = (2 - 1) = 1
m2 = - [(1 * 1) - (1 * 2)] = - (1 - 2) = 1
m3 = (1 * 1) - (2 * 2) = (1 - 4) = - 3
m4 = - [(1 * 1) - (1 * 2)] = - (1 - 2) = 1
m5 = (1 * 1) - (2 * 2) = (1 - 4) = - 3
m6 = - [(1 * 1) - (1 * 2)] = - (1 - 2) = 1
m7 = (1 * 1) - (2 * 2) = (1 - 4) = - 3
m8 = - [(1 * 1) - (1 * 2)] = - (1 - 2) = 1
m9 = (1 * 2) - (1 * 1) = 1


thus [1 1 -3]
M = [1 -3 1]
[-3 1 1]

now take transpose of matrix M, i.e. N

[1 1 -3]
N = [1 -3 1]
[-3 1 1]
matrix N is known as adjA i.e adjoint of A
thus
[1 1 -3]
adj A = [1 -3 1]
[-3 1 1]

now
inverse of matrix A let P is

[1 1 -3]
P = (1/ - 4) * [1 -3 1]
[-3 1 1]



thus [-1/4 -1/4 3/4]
P = [-1/4 3/4 -1/4]
[3/4 -1/4 -1/4]


to check
verify A * P = 1.



ab aa gaya samajh mein.
bye
take care.

Answer 2

Inverse of Matrix :
Let A be a non-singular matrix. If there exists a square matrix B such that AB = I (identity matrix) then B is called inverse of matrix A and is denoted as A-1.
i.e AA-1 = I
Example: Matrix A Matrix B = Identity (I)
1 3 1 * 2 9 -5 = 1 0 0
1 1 2 0 -2 1 0 1 0
2 3 4 -1 -3 2 0 0 1


In the above example Matrix A when multiplied by Matrix B gives an identity matrix. So we can call B as an inverse of A or A as an inverse of B.

Inverse of Matrix Tutorial:

In order to calculate the inverse of matrix we have to do the following steps
a) Find determinant of A (|A|)
b) Find adjoint of A (adj A)
c) Calculate the inverse using the formula.
A-1 = adjoint A / |A|

Answer 3

What you have to do is put it side by side with the identity matrix, and change the matrix to reduced row echelon form.

For instance, suppose you wanted to find the inverse of
[2 -1]
[0 3]

Put this in a new matrix, side by side with the identity matrix.

[2 -1 | 1 0]
[0 3 | 0 1]

Now, reduce the matrix on the left to the identity matrix using proper methods.

R1 -> 1/2 R1

[1 -1/2 | 1/2 0]
[0 3 | 0 1]

R2 -> 1/3 R2

[1 -1/2 | 1/2 0]
[0 1 | 0 1/3]

R1 -> R1 + (1/2) R2

[1 0 | 1/2 1/6]
[0 1 | 0 1/3]

Therefore, the inverse of

[2 -1]
[0 3]

is

[1/2 1/6]
[0 1/3]

How do we verify this? Through matrix multiplication.

[2 -1] [1/2 1/6]
[0 3] [0 1/3]

This equals

[1 - 0 1/3 - 1/3]
[0 + 0 0 + 3(1/3)]

which equals
[1 0]
[0 1]

Which means it's definitely the inverse.

You do the exact same thing with a 3 x 3 matrix, but instead you would of course put it side by side with
[1 0 0]
[0 1 0]
[0 0 1]

Answer 4

Calculate the determinant of the matrix. Then find out the adjoint of the matrix.Divide the adjoint of the matrix by the determinant.This gives us the inverse of the matrix.Inverse of a matrix cannot be found out if the determinant of the matrix is 0.

Answer 5

AX=B
X=A^-1 *B
det |A| must not be zero

You can calculate an inverse matrix using EXCEL
or a software (a computer program) .

Answer 6

1.find the determinant value of tat matrix
2.find the adjoint of tat matrix
3.divide the determinant value by the adjoint matrix
that is
inverse of A=|A|/adj(A)