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, A11 is the determinant of A excluding the first row and column. A12 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:

 

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