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Let's say we have the square n x n matrix A composed of real numbers. By "square", we mean it has the same numberof rows as columns.

A a 1 1 a 1 n a n 1 a n n

The subscripts of the real numbers in this matrix denote the row and column numbers, respectively (i.e. a 1 2 holds the position at the intersection of the first row and the second column).

We will denote the inverse of this matrix as A . A matrix inverse has the property that when it is multiplied by theoriginal matrix (on the left or on the right), the result will be the identity matrix.

A A A A I

To compute the inverse of A , two steps are required. Both involve taking determinants ( A ) of matrices. The first step is to find the adjoint ( A ) of the matrix A. It is computed as follows:

A 1 1 1 n n 1 n n
i j -1 i j A ij

where A ij is the n 1 x n 1 matrix obtained from A by eliminating its i -th column and j -th row . Note that we are not eliminating the i -th row and j -th column as you might expect.

To finish the process of determining the inverse, simply divide the adjoint by the determinant of the original matrix A .

A 1 A A

A a b c d

Find the inverse of the above matrix.

The first step is to compute the terms in the adjoint matrix:

1 1 -1 2 d
1 2 -1 3 b
2 1 -1 3 c
2 2 -1 4 a

Therefore,

A d b c a

We then compute the determinant to be a d b c . Dividing through by this quantity yields the inverse of A :

A 1 a d b c d b c a

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Source:  OpenStax, State space systems. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10143/1.3
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