Question

16) If A and B are square matrices of order n, then prove that A and B will commute
if and only if A-X and B-W commute for every scalar.​

Answer 1

$\huge \pink \bigstar \mathbb \color{cyan}A \color{pink}N \color{orange}S \color{blue}W \color{green}E \color{gray}R \color{purple} \leadsto$

• We can get the orthogonal matrix if the given matrix should be a square matrix.
• The orthogonal matrix has all real elements in it.
• All identity matrices are an orthogonal matrix.
• The product of two orthogonal matrices is also an orthogonal matrix.
• The collection of the orthogonal matrix of order n x n, in a group, is called an orthogonal group and is denoted by ‘O’.
• The transpose of the orthogonal matrix is also orthogonal. Thus, if matrix A is orthogonal, then is AT is also an orthogonal matrix.
• In the same way, the inverse of the orthogonal matrix, which is A-1 is also an orthogonal matrix.
• The determinant of the orthogonal matrix has a value of ±1.
• The eigenvalues of the orthogonal matrix also have a value as ±1, and its eigenvectors would also be orthogonal and real.