Select all the orthogonally diagonalizable matrices. \[ \begin{array}{l} {\left[\begin{array}{lll} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right]} \\ {\left[\begin{array}{lll} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]} \\ {\left[\begin{array}{lll} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]} \\ {\left[\begin{array}{lll} 1 & 1 & 0 \\ 1 & 1 & 1 \\ 1 & 0 & 1 \end{array}\right]} \\ {\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]} \\ {\left[\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]} \end{array} \]
Solution
Step 1 ：Given a set of matrices, we are asked to find which of them are orthogonally diagonalizable.
Step 2 ：A matrix is orthogonally diagonalizable if and only if it is symmetric. A matrix is symmetric if it is equal to its transpose.
Step 3 ：Let's check which of the given matrices are symmetric.
Step 4 ：The symmetric matrices are \[\begin{array}{lll} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right], \[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right], and \[\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right].
Step 5 ：Thus, the orthogonally diagonalizable matrices are \[\begin{array}{lll} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right], \[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right], and \[\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right].
Step 6 ：Final Answer: \(\boxed{\begin{array}{lll} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right], \boxed{\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right], \boxed{\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right}}\)
