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Adjoint and inverse of a square matrix

Recall that a cofactor matrix C of a matrix A is the square matrix of the same order as A in which each element aij is replaced by its cofactor cij .

Example: For then the cofactor, C of A is

The adjoint matrix of [A] is written as Adj[A] and it can be obtained by obtaining the transpose of the cofactor matrix of [A]. The minor for element aijof matrix [A] is obtained by removing the ith row and jth column from [A]. We then calculate the determinant of the remaining matrix. The following equation shows the adjoint matrix of A, denoted by adj A, which is the transpose of its cofactor matrix

Commutativity can be proven to show that:A(adj A) = (adjA) A = |A| I

Example:

Consider a scalar k. The inverse is the reciprocal or division of 1 by the scalar.

Example:k=7 the inverse of k or k-1 = 1/k = 1/7

Division of matrices cannot defined because in some casesĀ AB = AC while B = C. Instead matrix inversion is used. The inverse of a square matrix, A, if it exists, is the unique matrix A-1,where:AA-1 = A-1A = Iand A(adj A) = (adjA) A = |A| Ithen,

Consider the product A[adj(A)]

The entry as the position (I, j) of A[adj(A)]

Consider a matrix BĀ similar to matrix A except that the j-th row is replaced by the i-th row of matrix A

Example: For 2×2 matrix, calculate the inverse:

Example:

Example:

The result can be verified using AA-1 =A-1A = I. Therefore, the determinant of a matrix must not be zero for the inverse to exist as there will not be a solution. Nonsingular matrices have non-zero determinants and singular matrices have zero determinants

Consider a simple 2 x 2 square matrix:

Multiplying gives:

  • aw + by = 1
  • ax + bz = 0
  • cw + dy = 0
  • cx + dz = 1

This can be shown as

Therefore,

Hence, for 2 x 2 matrix, we can write the inverse as:

  1. Exchange elements of main diagonal
  2. Change sign in elements off main diagonal
  3. Divide resulting matrix by the determinant

 

 

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