Recall that a cofactor matrix **C** of a matrix **A** is the square matrix of the same order as **A** in which each element a* _{ij}* is replaced by its cofactor c

*.*

_{ij}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 *a _{ij}*of matrix [A] is obtained by removing the

*i*th row and

*j*th 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**) = (adj**A**) **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**^{-1}**A** = **I**and **A**(adj **A**) = (adj**A**) **A** = |**A**| **I**then,

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**^{-1}**A** = **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:

- Exchange elements of main diagonal
- Change sign in elements off main diagonal
- Divide resulting matrix by the determinant