Special matrices

     The 2×2

has the property that, for any 2×2 matrix A,

In other words, multiplication by I (either pre-multiplication or post-multiplication) leaves the elements of A unchanged. I is called the identity matrix and it is analogous to the real number 1 in ordinary multiplication.

The inverse of a matrix

In matrix arithmetic we thus require, for a given matrix A, the matrix B for which, AB=BA=I. B is denoted by A⁻¹ and is called the inverse matrix of A, giving AA⁻¹=A⁻¹A=I

The inverse of a matrix is given by; 

Where, det= determinant and adj= adjacent.

Finding the determinant of a 2x2 matrix

For any matrix

, the determinant is given by detA=(ad-bc).

Example;

Given the matrix 

, find its determinant;

Solution

DetA = [(2x5) – (1x4)] = 10 – 4= 6

DetA = 6

Finding the adjacent of a 2x2 matrix

For any 2x2 matrix  , adjacent of matrix A is given by;

Example

Given the matrix  , find adjacent A 

Solution 

Calculating the inverse of a matrix

Given the matrix A= 

Solution;

Step 1: DetA = [(2x5) – (1x4)] = 10 – 4= 6 

              DetA = 6