Matrix Inverse

An n x n matrix, times its MatrixInverse, equals the IdentityMatrix?.

 B  *  B^(-1) = I

The IdentityMatrix? is an n x n matrix with 1's along the main diagonal, and 0's everywhere else. When an n x n matrix A is multiplied by an n x n identity matrix I, the answer is A. (Just like when a scalar a is multiplied by 1, the answer is a.)

Some square matrices do not have inverses.

The MatrixInverse = the AdjointMatrix divided by the MatrixDeterminant. Both the AdjointMatrix and the MatrixDeterminant are calculated recursively. For 2 x 2 matrices, these are easy to calculate. As the matrices get larger, these calculations become very tedious very quickly.

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For a 2 x 2 matrix:

          [ b11  b12 ]
      B = [          ]
          [ b21  b22 ]

The adjoint matrix is:

          [ b22 -b12 ]
 adj(B) = [          ]
          [-b21  b11 ]

The determinant is:

 det(B) = b11 * b22  -  b12 * b21

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So, for the example on the MatrixFactoring page, we get:

          [ 0.795     8.805 ]   
      B = [                 ] 
          [ 0.205    -7.805 ]

          [-7.805    -8.805 ]
 adj(B) = [                 ]
          [-0.205    0.795  ]

 det(B) = (0.795)*(-7.805) - (0.205)*(8.805)
        = -8.01

          [ 0.9744   1.09925]
 B^(-1) = [                 ]
          [ 0.0256  -0.09925]

 [ 0.795  8.805 ]   [ 0.9744  1.09925]     [ 1  0 ]
 [              ] * [                ]  =  [      ]
 [ 0.205 -7.805 ]   [ 0.0256 -0.09925]     [ 0  1 ]

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The general case of finding a MatrixInverse can be solved by augmenting an IdentityMatrix? and using GaussianElimination

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CategoryMath