# Derived Forms of Matrices

### Matrix Transpose

In a matrix, if we change the position of the elements so that the rows become columns and the columns become rows, we get the transpose of the initial matrix.

And a numerical example:

### Adjugate or Adjoint Matrix

Consider the following square matrix **A**:

The adjugate of **A** can be acquired if we create the transpose of a matrix that contains as elements, the determinants of matrix **A**, excluding for each determinant the corresponding row and column of matrix **A**.

For the matrix above, A_{11} is the determinant of **A** excluding the first row and column. A_{12} is the determinant of **A** excluding the first row and second column e.t.c.

For a numerical example we will calculate the adjugate of the following matrix.

The determinant elements are:

The adjugate of matrix **A** is:

### Inverse Matrix

The inverse of a matrix can be found using the following formula:

To calculate the inverse of matrix **A**:

The determinant of **A** is:

From the previous paragraph we calculated adj(**A**), so:

### More about Matrices