Explain the concept of Inverse Matrices?

asked 2020-12-28
Explain the concept of Inverse Matrices?

Expert Answers (1)

Step 1
The inverse of a matrix X is a matrix such that when we multiply the matrix X by its inverse the result is the identity matrix. The symbol that we use to denote for the inverse matrix for \(X \text{ is } X^{-1}\)
That is \(XX^{-1}=X^{-1}X=1\)
There are two conditions for a matrix to have inverse.
(1) The matrix should be a square matrix.
(2) The determinant of the matrix should be non-zero \(|X| \neq 0\)
Step 2
The matrix which are not square matrix do not have the inverse.
A matrix which has inverse it also known as invertible matrix.
Here we can see as an example how to find inverse of a matrix
Let us consider a matrix
\(X=\begin{pmatrix}a & b \\c & d \end{pmatrix}\)
\(|X|=(ac-bd)\neq 0\)
\(X^{-1}=\frac{1}{|X|}\begin{pmatrix}d & -b \\-c & a \end{pmatrix}\)
\(X^{-1}=\frac{1}{(ac-bd)}\begin{pmatrix}d & -b \\-c & a \end{pmatrix}\)
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