Question

Define Invertible Matrices. Give an example.

Matrices
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asked 2020-12-28
Define Invertible Matrices. Give an example.

Answers (1)

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
\(|A|=\begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix}\)
\(=1 \cdot (3) - 2 \cdot (-1)\)
\(=3+2\)
\(=5\)
\(|A| \neq 0 \rightarrow A \text{ is invertible.}\)
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