How do you go about getting the inverse of a matrix?

Answer 1

Pre-Answer Disclaimer: I'm assuming a very specific type of matrix in my result. The formula I provide for A^(-1) works in fields outside the real numbers as well, but I am solely using real values for the sake of simplicity. Also, I provide a 2 x 2 example, but the formula works for the n x n case as well (in fact, it was developed for larger matrices). Message me (or ask "how do you go about getting the cofactors of a matrix") for more info on this topic or others.

Given any square matrix A with a nonzero determinant detA, the inverse of A is denoted A^(-1) (for lack of proper typesetting) and is defined as follows:
A^(-1) = (1/detA)*(C_i,j)^T, where (C_i,j) is the matrix of the cofactors (signed minors) of A=(a_i,j) and "^T" denotes the transpose of this matrix.
For example, (again for lack of better typesetting...hint hint yahoo : ] ) suppose
A = ( 1 , 2 ; 3 , 4 ), where commas separate consecutive values within a row and semi-colons denote changes in row.
detA = 1*4 - 2*3 = 4 - 6 = -2 <> 0,
and therefore A is invertible.
M_1,1 = 4
M_1,2 = 3
M_2,1 = 2
M_2,2 = 1.
(Remember, the minor of the entry in the ith row and jth column is equal to the determinant of the matrix that is generated by "crossing out" the ith row and jth column.)
Thus, applying the "checkerboard" pattern that converts minors to cofactors, we have that
C_1,1 = 4
C_1,2 = -3
C_2,1 = -2
C_2,2 = 1.
Therefore
C_i,j= ( 4 , -3 ; -2 , 1 ),
and
(C_i,j)^T = ( 4 , -2 ; -3 , 1 ).
Substituting this value into the equation for A^(-1) along with (1/detA)= -(1/2), we have that
A^(-1) = (1/detA)*(C_i,j)^T = -(1/2)( 4 , -2 ; -3 , 1 )
= ( -2 , 1 ; 3/2 , -1/2).
It's always easy to check your work, since the product A*A^(-1) must always equal 1. Here we see that
A*A^(-1) = ( 1 , 2 ; 3 , 4 )*( -2 , 1 ; 3/2 , -1/2)
= ( 1(-2) + 2(3/2) , 1*1 + 2(-1/2) ; 3(-2) + 4(3/2) , 3*1 + 4(-1/2) )
= ( 1 , 0 ; 0 , 1 ) = I_2, just as we wish.

Answer 2

dont make it negative! find the determinate and then the inverse is the recipical of that

Answer 3

search on google or wikipedia or better books
do you expect to learn from ya??!?

Answer 4

for a 2x2 matrix

|a...b| the inverse is 1/(ad-bc)|d -b|
|c...d|........................................................... | -c a|

for 3x3 you need to find that in a book or your TI calc.

Answer 5

negative reciprocal.. i have no clue lol