Question

A square matrix ($a_{ij}$) in which $a_{ij}=0$ for $i\neq j$ and $a_{ij}=k$(constant) for i=j is
A) Unit matrix
B) Scalar matrix
C) Null matrix
D) None

Answer 1

Hint: In this question it is given that a square matrix ($a_{ij}$) in which $a_{ij}=0$ for $i\neq j$ and $a_{ij}=k$(constant) for i=j, we have to find the type of this matrix. So for this we have to construct the matrix.
Complete step-by-step solution:
Let us consider that the order of this square matrix is $n\times n$, and the given values of the elements is $a_{ij}=0$ for $i\neq j$ and $a_{ij}=k$(constant) for i=j.
So from the above condition we can write,
$a_{11}=a_{22}=a_{33}=\cdots \cdots =a_{nn}=k$
And apart from these elements, all the other elements of the matrix are zero.
So we can write the given matrix as,
$\begin{bmatrix}k&0&0&\cdots &0&0\\\ 0&k;&0&\cdots &0&0\\\ 0&0&k;&\cdots &0&0\\\ \vdots &\vdots &\vdots &\ddots &\vdots &\vdots \\\ 0&0&0&\cdots &k;&0\\\ 0&0&0&\cdots &0&k;\end{bmatrix}$

In the given matrix all the elements other than the diagonal are zero and diagonal elements are equal to K , so as we know that "a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero " and "A diagonal matrix with all its main diagonal entries equal is a scalar matrix"
So we can say that the correct option is option B.

Note: To Solve these types of questions you need to know that $a_{ij}$ is the element of a matrix where ij defines $i^{th}$ row and $j^{th}$ column, and also you need to know that diagonal matrix with all its main diagonal entries equal is called a scalar matrix.