Matrices are used to represent data in rows and columns
A matrix which is its own inverse is the identity matrix.
An 2-by-2 identity matrix is represented as:
\(\mathbf{A = \left[\begin{array}{cc}1&0\\0&1\end{array}\right] }\)
Calculate the determinant (d)
\(\mathbf{d = 1 \times 1 - 0 \times 0}\)
\(\mathbf{d = 1}\)
So, the inverse matrix is:
\(\mathbf{A^{-1}= \frac{1}{d} A^T}\)
This gives
\(\mathbf{A^{-1}= \frac{1}{1} A^T}\)
\(\mathbf{A^{-1}= A^T}\)
Transpose the matrix.
So, we have
\(\mathbf{A^T = \left[\begin{array}{cc}1&0\\0&1\end{array}\right] }\)
By comparison
\(\mathbf{A=A^T = \left[\begin{array}{cc}1&0\\0&1\end{array}\right] }\)
Hence, a matrix which is its own inverse is the identity matrix.
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