The Inverse of a 2 by 2 Matrix

Topic Content:

A 2 by 2 matrix is also referred to as a square matrix. The inverse of a 2 × 2 matrix , say A, is a matrix of the same order denoted by A^-1 , such that:

AA^-1 = A^-1 A = I

where I is the identity matrix of order 2×2.

i.e. I = \(\scriptsize \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)

If you multiply a 2×2 matrix A by I we should get A. This is similar to the multiplication of integers, e.g 2 × 1 = 2.

The inverse of a matrix is another matrix that, when multiplied by the given matrix, yields the multiplicative identity.

Example 4.3.1:

Given matrix A, show that AI = A

A = \(\scriptsize \begin{pmatrix} 3 & 4 \\ -2 & 5 \end{pmatrix}\)

Solution

AI = \(\scriptsize \begin{pmatrix} 3 & 4 \\ -2 & 5 \end{pmatrix}\scriptsize \: \times \: I\)

Where I = \(\scriptsize \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)

∴ AI = \(\scriptsize \begin{pmatrix} 3 & 4 \\ -2 & 5 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)

AI = \(\scriptsize \begin{pmatrix} 3 \: \times \: 1 \: + \: 4 \: \times \: 0 & 3 \: \times \: 0 \: + \: 4 \: \times \: 1 \\ -2\: \times \: 1 \: + \: 5 \: \times \: 0 & -2\: \times \: 0 \: + \: 5 \: \times \: 1 \end{pmatrix}\)

AI = A = \(\scriptsize \begin{pmatrix} 3 & 4 \\ -2 & 5 \end{pmatrix}\)

Finding the Inverse of a 2×2 Matrix Using Matrix Multiplication:

Example 4.3.2: