A diagonal matrix is a square matrix in which all the elements outside the main diagonal are zero. The elements on the main diagonal can be any value, including zero.
We need to find the condition that ensures all elements outside the main diagonal are zero.
Option (A) states: $b_{ij} = 0$, when $i = j$. This means the elements on the main diagonal are zero. This is a special case of a diagonal matrix, but not the general definition.
Option (B) states: $b_{ij} = 1$, when $i = j$. This means the elements on the main diagonal are one. This describes an identity matrix, which is a special case of a diagonal matrix, but not the general definition.
Option (C) states: $b_{ij} = 1$, when $i \neq j$. This means all elements outside the main diagonal are one, which is not a diagonal matrix.
Option (D) states: $b_{ij} = 0$, when $i \neq j$. This means all elements outside the main diagonal are zero, which is the definition of a diagonal matrix.
Final Answer: $b_{ij} = 0$, when $i \neq j$
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