Identify the non-diagonal matrix from the given options.

A. $ \left[ \begin{gathered}
  {\text{1 0 0}} \\
  {\text{0 1 0}} \\
  {\text{0 0 1}} \\
\end{gathered} \right] $
B. $ \left[ \begin{gathered}
  {\text{1 0 0}} \\
  {\text{0 2 0}} \\
  {\text{0 9 0}} \\
\end{gathered} \right] $
C. $ \left[ \begin{gathered}
  {\text{1 0 0}} \\
  {\text{0 4 0}} \\
  {\text{0 0 1}} \\
\end{gathered} \right] $
D. $ \left[ \begin{gathered}
  {\text{1 0 0}} \\
  {\text{0 4 0}} \\
  {\text{0 0 - 1}} \\
\end{gathered} \right] $

Hint: Before attempting this question, one should have prior knowledge about the types of matrix and also remember that in diagonal matrix elements other than diagonal elements are zero, using this information will help you to approach towards the solution of the question.

Complete step-by-step answer:
According to the given information we know that diagonal matrix is those matrices in which elements other than diagonal elements are zero
As in the above information we have 4 different matrices therefore to find the non-diagonal matrix we have to identify if there are any elements other than diagonal elements that are non-zero.
For matrix: $ \left[ \begin{gathered}
  {\text{1 0 0}} \\
  {\text{0 1 0}} \\
  {\text{0 0 1}} \\
\end{gathered} \right] $
As in the above the non-zero elements are only arranged in diagonal and the rest of the elements are zero
Therefore, the above matrix is a diagonal matrix
For matrix: $ \left[ \begin{gathered}
  {\text{1 0 0}} \\
  {\text{0 2 0}} \\
  {\text{0 9 0}} \\
\end{gathered} \right] $
As in the above matrix the zero elements are also present in diagonal
Therefore, the above matrix is a non-diagonal matrix
For matrix: $ \left[ \begin{gathered}
  {\text{1 0 0}} \\
  {\text{0 4 0}} \\
  {\text{0 0 1}} \\
\end{gathered} \right] $
As in the above the non-zero elements are only arranged in diagonal and the rest of the elements are zero
Therefore, the above matrix is a diagonal matrix
For matrix: $ \left[ \begin{gathered}
  {\text{1 0 0}} \\
  {\text{0 4 0}} \\
  {\text{0 0 - 1}} \\
\end{gathered} \right] $
As in the above the non-zero elements are only arranged in diagonal and the rest of the elements are zero
Therefore, the above matrix is a diagonal matrix
So, we can say that only matrix i.e. $ \left[ \begin{gathered}
  {\text{1 0 0}} \\
  {\text{0 2 0}} \\
  {\text{0 9 0}} \\
\end{gathered} \right] $ is a non-diagonal matrix
So, the correct answer is “Option B”.

Note: In the above solution we came across the term “matrix” which can be explained as a method of arranging of numbers, symbols and equations into rows and columns in such a way that the arrangement make a rectangular arrangement there are different types of matrix such as unit matrix, square matrix row matrix, column matrix, null matrix, etc.