Question

# Define Invertible Matrices. Give an example.

Matrices
Define Invertible Matrices. Give an example.

2020-12-29
Step 1
An $$n \times n$$ square matrix A is called Invertible Matrices, if there exists an n x n square matrix B such that AB = BA = I.
$$AB=BA=I_n$$
$$A_{n \times n} , B_{n \times n} \text{ and } I_n \text{ is identity matrix of } n \times n$$
Step 2
In other words, determinant of A is non - zero, then matric A is invertible. $$det(A)=|A| \neq 0$$
Step 3
For example, let us assume
$$A=\begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix}$$
Step 4
So, the determinant of A is
$$|A|=\begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix}$$
$$=1 \cdot (3) - 2 \cdot (-1)$$
$$=3+2$$
$$=5$$
$$|A| \neq 0 \rightarrow A \text{ is invertible.}$$