Matrix Inverse - Theory

1. Find det(M), the determinant of the Matrix M. The determinant will usually show up in the denominator of the inverse. If the determinant is zero, the matrix won't have an inverse.
   
   
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2. 
   
   Find M^T, the transpose of the matrix. Transposing means reflecting the matrix about the main diagonal, or equivalently, swapping the (i,j)th element and the (j,i)th.
   
   
3.  Find the determinant of each of the 2x2 minor matrices.Find the determinant of each of the 2x2 minor matrices.
   
   
4. 
   
   Represent these as a matrix of cofactors as shown, and multiply each term by the sign indicated. The result of these steps is the adjugate matrix (sometimes also called the adjoint), notated Adj(M).
   
   
5. 
   
   Find the inverse by dividing the adjugate found in the previous step by the determinate from the first step.
   
   

• Note that this same method can be applied to a matrix containing variables or unknowns, for example an algebraic matrix, M, and its inverse, M^-1.
  
  
  
• Write down all your steps as it is extremely difficult to inverse a 3x3 matrix in your head
• Computer programs exist that work out the inverses of matrices for you, up to and including the size of 30x30 matrices
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• The adjugate matrix is the transpose of the matrix of co-factors, that is why we transpose the matrix in step 2, to find the a transposed matrix of co-factors.
• Check that is accurate by multiplying M by M^-1. You should be able verify that M*M^-1 = M^-1*M = I. I is the identity matrix, consisting of 1s along the main diagonal and 0s elsewhere. If not, you made an error somewhere.