Help with inverses and determinants of matrixes?

Question

How do you compute the inverse of a matrix using the determinant, I think it is called crammers’ rule.
any help would be much appreciated
Thanks

Answer 1

A^(-1) = ( 1/det(A) ) * adj(A)



Where det = determinant, and

adj = adjoint



There is a trick to getting the Adjoint of a matrix, almost similar to the determinant (except you are computing entries for a new matrix). You then transpose the result of doing so.



Example: Find the adjoint of

[1 0 1]

[2 2 5]

[-1 3 0]



Answer: Your matrix will be the result of finding the determinant after deleting the i-th row and j-th column for enter a[ij]. You will also negate the answer if (i + j) is odd. So you will have



[+ - +] T

[- + -]

[+ - +]



(transpose of the answer)



To find the entry at 1, 1, delete 1st row and 1st column, to get



[2 5]

[3 0]



And find the determinant of that. The result will be your first entry. The determinant of this 2 x 2 matrix is -15.



[-15 - +] T

[- + -]

[+ - +]



To find (1, 2), delete 1st row second column, leaving you with just

[2 5]

[-1 0]



For a determinant of 5. But since the entry is a negative, we negate the 5 to get -5.



[-15 -5 +] T

[- + -]

[+ - +]



If you keep doing this to find the entries, you will eventually get



[-15 -5 8] T

[3 1 -3]

[-2 -3 2]



And you want to transpose this.



[-15 3 -2]

[-5 1 -3]

[8 -3 2]



That means



adjoint of the given matrix is that.



With that said, we can now calculate the inverse of our original matrix,which we will call A.

A = [1 0 1]

. . . [2 2 5]

. . . [-1 3 0]





A^(-1) = ( 1/det(A) ) adj(A)



We evaluate the determimant.



det(A) = 1 * det[2 5] - 0 + 1 * det[2 2]

. . . . . . . . . . . .[3 0] . . . . . . . . [-1 3]



det(A) = 1 * (0 - 15) + 1 * (6 - (-2))

det(A) = 1(-15) + (6 + 2)

det(A) = -15 + 8

det(A) = -7



And we already know that

adj(A) = [-15 3 -2]

. . . . . . .[-5 1 -3]

. . . . . . .[8 -3 2]



So



A^(-1) = ( 1/(-7) ) [-15 3 -2]

. . . . . . . . . . . . .[-5 1 -3]

. . . . . . .. . . . . . .[8 -3 2]



A^(-1) = ( -1/7 ) [-15 3 -2]

. . . . . . . . . . . . .[-5 1 -3]

. . . . . . .. . . . . . .[8 -3 2]



A^(-1) = [ 15/7 -3/7 2/7 ]

. . . . . . [ 5/7 -1/7 3/7 ]

. . . . . . [ -8/7 3/7 -2/7 ]



Barring arithmetic errors, this should be the inverse.

To test, just multiply it with original matrix and see if you get identity.



[1 0 1][ 15/7 -3/7 2/7 ]

[2 2 5] [ 5/7 -1/7 3/7 ]

[-1 3 0] [ -8/7 3/7 -2/7 ]



[1 0 0]

[0 1 0]

[0 0 1]



Yup.

Answer 2

To find the inverse and determinants of matrices is work to be done very carefully and very minutely.



You can know to find inverse and determinant of matrices if you get some good book and practice time and again.



Cramer's Rule is a method to find the solution of system of equations in two or three or n variables.

Answer 3

The inverse is:

A^-1 = (1/det(A))(adj(A))



det(A) is the determinant of A

you divide every element of the matrix adj(A) by det(A)

adj(A) is the adjoint of A

The adjoint of a matrix is the transpose of its cofactor

and its cofactor is the matrix which each of its elements is the determinant of the minor of the previous matrix's element multiplied by the sign of the element.

Answer 4

formula is

A^-1=adj A/|A|

Inverse of A=adj A divided by determinant A. (a is a matrix)