Here you will learn what is involutory matrix with examples.
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Involutory Matrix
If \(A^2\) = I . the matrix A is said to be an involutory matrix, i.e. the square roots of the identity matrix (I) is involutory matrix.
Note : The determinant value of this matrix (A) is 1 or -1.
Example : Show that the matrix A = \(\begin{bmatrix} -5 & -8 &ย 0 \\ย 3 & 5 & 0 \\ย 1 & 2 & -1 \end{bmatrix}\) is involutory.
Solution : We have,
A = \(\begin{bmatrix} -5 & -8 &ย 0 \\ย 3 & 5 & 0 \\ย 1 & 2 & -1 \end{bmatrix}\)
Now we find, \(A^2\) = A . A
\(\implies\) \(A^2\) = \(\begin{bmatrix} -5 & -8 &ย 0 \\ย 3 & 5 & 0 \\ย 1 & 2 & -1 \end{bmatrix}\) \(\times\) \(\begin{bmatrix} -5 & -8 &ย 0 \\ย 3 & 5 & 0 \\ย 1 & 2 & -1 \end{bmatrix}\)
= \(\begin{bmatrix} 1 & 0 &ย 0 \\ย 0 & 1 & 0 \\ย 0 & 0 & 1 \end{bmatrix}\)
Hence, the given matrix A is involutary.
Example : Show that the sqare matrix A is involutary, iff (I โ A) (I + A) = O
Solution : Let A is the involutary matrix,
Then, \(A^2\) = I
(I โ A) (I + A) = \(I^2\) + IA โ AI โ \(A^2\)
= I + A โ A โ \(A^2\)ย
= I โ \(A^2\) = O
Conversely, let (I โ A) (I + A) = O
\(\implies\) \(I^2\) + IA โ AI โ \(A^2\) = O
\(\implies\) I + A โ A โ \(A^2\) = O
= I โ \(A^2\) = O
Hence, the given matrix A is involutary.
