Question

Define a diagonal matrix.

Answer 1

Hint: The word diagonal itself gives an idea about the matrix which refers to having to do something with the diagonal elements of the matrix.
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices.
It is only applicable or available in square matrices. Where the number of rows is equal to the number of columns of the matrices.

Complete step-by-step answer:
Diagonal matrices are those with 0 elements everywhere but along the diagonal in only the square matrices:
$M = \left( {\begin{array}{*{20}{c}}
  {{a_{11}}}& \ldots &{{a_{1n}}} \\
   \vdots &{{a_{2,2}}}& \vdots \\
  {{a_{m1}}}& \cdots &{{a_{mn}}}
\end{array}} \right)$

Diagonal matrices have some properties that can be usefully learned for matrix addition, multiplication and furthermore operations:
I.If $A$ and $B$ are diagonal matrices, then $C = AB$ is a diagonal matrix. Further, C can be computed more easily than naively doing a full matrix multiplication: ${c_{ii}} = {a_{ii}}{b_{ii}}$, and all other entries are 0.
II.Multiplication of diagonal matrices is commutative: if $A$ and $B$ are diagonal matrices, then $C = AB = BA$ in which both the end values are equal which can be easily calculated..
III.If $A$ is diagonal matrix, and $B$ is a general matrix, and $C = AB$ , then the ith row of $C$ is ${a_{ii}}$ times the ith row of B; if $C = BA$, then the ith column of matrix $C$ is ${a_{ii}}$times the ith column of $B$ matrix.
So, the correct answer is “Option C”.

Note: Do not mistake that diagonal matrix is only available or applicable in square matrices it is not applicable for other matrices where the number of rows and columns are not the same.