Question

Choose the correct answer:
A) Every scalar matrix is an identity matrix
B) Every identity matrix is a scalar matrix
C) Every diagonal matrix is an identity matrix
D) A square matrix whose element is 1 is an identity matrix.

Answer 1

Hint: Here the definitions of different kinds of matrices- Scalar, Identity, Diagonal and square need to be considered very carefully to choose the correct statement. All the matrices are very similar to each other with very little difference but not one of them is the same as the other.

Complete step by step answer:
First of all, we need to understand the different matrices mentioned in the different statements.
1) Now, a scalar matrix is a matrix which is a scalar multiple of a unit matrix of any order. For example, if we consider a 2 x 2 matrix then a scalar matrix will be written as:
$\left[\begin{array}{cc}a& 0\\ 0& a\end{array}\right]=a\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$where a, is a scalar quantity. So the scalar multiple will be outside and the matrix will have the diagonal elements as 1 and the rest of the elements will be 0.

2) For an identity matrix, all elements except the ones diagonally will be 0 and the diagonal elements will have to be 1.
For example, an identity matrix or order 3 x 3 will be given as:
$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

3)Diagonal matrix of any order is a matrix in which except for the diagonal elements all the other elements are 0 and the diagonal elements can be any numbers other than 1 as well.
For example, a diagonal matrix of order 4 x 4 will be:
$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& -2& 0& 0\\ 0& 0& 4& 0\\ 0& 0& 0& 5\end{array}\right]$

4) Square matrix is a matrix in which the number of rows and columns are the same.
Hence its order is nxn. For example, a square matrix of order 2 x 2 will have 2 rows and 2 columns. Similarly, a square matrix of order 3 x 3 will have 3 rows and 3 columns.
Example of square matrix of order 2 x 2:
$\left[\begin{array}{cc}2& -1\\ 4& 7\end{array}\right]$

Now, from the definitions of the different matrices we can conclude that a scalar matrix is an identity matrix only if the scalar multiplying factor is 1. Hence every scalar matrix is not an identity matrix.
Every identity matrix is a scalar matrix because the scalar multiplying factor is 1 here.
Every diagonal matrix cannot be an identity matrix because not all diagonal elements will be 1 in a diagonal matrix.
A square matrix may or may not have the elements other than the diagonal ones as 0 and the diagonal elements may or may not be 1. So a square matrix cannot be an identity matrix.

Hence, the correct answer is Option B.

Note: The order of a matrix is given as $m\times n$ where m, are the number of rows and n are the number of columns. In a square matrix, $m=n$ such that the order becomes $n\times n$. A unit matrix is such a square matrix whose diagonal elements are 1 and the rest of the elements are 0. A scalar matrix is a unit matrix multiplied with a scalar quantity. A diagonal matrix is a square matrix, in which except the diagonal elements all other elements are 0, while the diagonal elements are not 1 necessarily.