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How does one find the inverse of a non-singular 3x3 matrix A?

(i) Find the DETERMINANT of A:

1. Multiply the (1,1)th element by the determinant of the 2x2 matrix that remains when the row and the column containing the element are crossed out.

2. Repeat for the elements (1,2)th and (1,3)th elements.

3. Alternately add and subtract the results.

(ii) Find the ADJUGATE of A (also known as the classical adjoint of A):

1. Form M, the matrix of minors of A, where the minor of an element of a 3x3 matrix is the determinant of the 2x2 matrix that remains when the row and the column containing the element are crossed out.

2. Form C, the matrix of cofactors, by changing the signs of alternating elements of M starting with the top left element as positive.

3. Form adj(A) = C^T, the transpose of the matrix of cofactors, by reflecting the matrix about the main diagonal (i.e. swapping the (i,j)th element with the (j,i)th element.)

(iii) Find the INVERSE of A, given by

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

where det(A) us the determinant of A and adj(A) is the adjugate of A.

NB: The same method can also be applied to larger square matrices but the process is long so instead we use the Guass-Jordon Method of Elimination, which is also applicable to 3x3 matrices.

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