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.