Question

Consider following statements:
1) Every zero matrix is a square matrix.
2) A matrix has a numerical value.
3) A unit matrix is a diagonal matrix.
Which of the above statements is/are correct?
A) 2 only
B) 3 only
C) 2 and 3 only

Answer 1

Hint:
First we will get what is zero matrix, square matrix, unit matrix and diagonal matrix. Then we will see which statement is true according to their definitions.

Complete step by step solution:
Zero matrix: It is the matrix with all elements zero. It is also called null matrix. But it is not necessarily a square matrix. It can be of dimensions \[m \times n\] or \[n \times n\]. Where m is the number of rows and m is the number of columns.
\[\left( {\begin{array}{*{20}{c}}
  0&0 \\
  0&0
\end{array}} \right)\] or \[\left( {\begin{array}{*{20}{c}}
  0 \\
  0 \\
  0
\end{array}\begin{array}{*{20}{c}}
  0 \\
  0 \\
  0
\end{array}} \right)\] both are zero matrices.

Square matrix: It is the matrix having the same number of rows and columns. That is having dimensions \[n \times n\].
Unit matrix: It is the matrix whose diagonal elements are 1 and rest other elements are zero. It is also called identity matrix.
\[\left( {\begin{array}{*{20}{c}}
  1&0 \\
  0&1
\end{array}} \right)\]
Diagonal matrix: It is the matrix with only diagonal elements present, the rest other are zero.
From these definitions it is clear that statement 1 is false but statement 3 is true.
Now about statement 2 we can say that a matrix is not having value; whereas it is a determinant that has a value. So statement 2 is also false.

Thus option B is the correct option.

Note:
Students note that matrices need not to be square always. Their dimensions may vary. Also note that every unit matrix is a diagonal matrix but not the reverse case is true always. We can have mathematical operations with matrices.