Adjoint and Inverse of a Matrix - GeeksforGeeks

Adjoint and Inverse of a Matrix - GeeksforGeeks

Adjoint and Inverse of a Matrix

Given a square matrix, find adjoint and inverse of the matrix.

We strongly recommend you to refer below as a prerequisite of this.
Determinant of a Matrix

What is Adjoint?
Adjoint (or Adjugate) of a matrix is the matrix obtained by taking transpose of the cofactor matrix of a given square matrix is called its Adjoint or Adjugate matrix. The Adjoint of any square matrix 'A' (say) is represented as Adj(A).

Example:

Below example and explanation are taken from here.  5  -2  2  7  1   0  0  3  -3  1  5  0  3  -1 -9  4    For instance, the cofactor of the top left corner '5' is   + |0   0   3|  ...|1   5   0| = 3(1 * -9 - (-1) * 5) = -12.  ...|-1 -9   4|  (The minor matrix is formed by deleting the row    and column of the given entry.)    As another sample, the cofactor of the top row corner '-2' is    -|1   0  3|  ...|-3  5  0| = - [1 (20 - 0) - 0 + 3 (27 - 15)] = -56.  ...|3  -9  4|    Proceeding like this, we obtain the matrix  [-12  -56   4   4]  [76   208   4   4]  [-60  -82  -2  20]  [-36  -58  -10 12]    Finally, to get the adjoint, just take the previous  matrix's transpose:  [-12   76 -60  -36]  [-56  208 -82  -58]  [4     4   -2  -10]  [4     4   20   12] 

Important properties:

• Product of a square matrix A with its adjoint yields a diagonal matrix, where each diagonal entry is equal to determinant of A.
  i.e.,

  A.adj(A) = det(A).I     I  => Identity matrix of same order as of A.  det(A) => Determinant value of A 

• A non zero square matrix 'A' of order n is said to be invertible if there exists a unique square matrix 'B' of order n such that,

     A.B = B.A = I  The matrix 'B' is said to be inverse of 'A'.  i.e.,  B = A-1

How to find Adjoint?
We follow definition given above.

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