Question

Which one of the following matrices is an elementary matrix?
A) \[\left( {\begin{array}{*{20}{c}}
  1&0&0 \\
  0&0&0 \\
  0&0&1
\end{array}} \right)\]
B) \[\left( {\begin{array}{*{20}{c}}
  1&5&0 \\
  0&1&0 \\
  0&0&1
\end{array}} \right)\]
C) \[\left( {\begin{array}{*{20}{c}}
  0&2&0 \\
  1&0&0 \\
  0&0&1
\end{array}} \right)\]
D) \[\left( {\begin{array}{*{20}{c}}
  1&0&0 \\
  0&1&0 \\
  0&5&2
\end{array}} \right)\]

Answer 1

Hint: A matrix is an elementary matrix if that matrix comes from an identity matrix. By applying one elementary row or column operation, an elementary matrix is obtained. We can check it by applying matrix operation on a $3 \times 3$ identity matrix.

Complete step by step solution:
An identity matrix is a square matrix whose diagonal elements are equal to 1 while the rest of the other elements are zero. E.g.
\[ \Rightarrow A = \left( {\begin{array}{*{20}{c}}
  1&0&0 \\
  0&1&0 \\
  0&0&1
\end{array}} \right)\], where ${a_{ij}}$represent the elements of a matrix & ‘i’ denotes the row number and ‘j’ denotes the column number.
A is the identity matrix and you can see that the diagonal elements are 1 while the rest of the other elements are zero.
By applying one elementary operation either row-wise or column-wise, we can tell which one of the matrices given in the option is an elementary matrix.
First of all, check option A. How can you check that it is an elementary matrix or not? By taking an $3 \times 3$ identity matrix of
\[ \Rightarrow A = \left( {\begin{array}{*{20}{c}}
  1&0&0 \\
  0&1&0 \\
  0&0&1
\end{array}} \right) \leftrightarrow \left( {\begin{array}{*{20}{c}}
  1&0&0 \\
  0&0&0 \\
  0&0&1
\end{array}} \right)\]
Now in the option A, ${a_{22}} = 0$ so this is not an elementary matrix. As if we apply elementary operation then the value of ${a_{22}} = k$, k can be any constant but not zero.
Check option B, by applying same previous method
\[ \Rightarrow A = \left( {\begin{array}{*{20}{c}}
  1&0&0 \\
  0&1&0 \\
  0&0&1
\end{array}} \right) \leftrightarrow \left( {\begin{array}{*{20}{c}}
  1&5&0 \\
  0&1&0 \\
  0&0&1
\end{array}} \right)\], ${R_1} \leftrightarrow {R_1} + 5{R_{_2}}$
Now in the option B, ${a_{11}} = 1,{a_{22}} = 1\& {a_{33}} = 1$, so this can be an elementary matrix. As if we apply elementary operation i.e. ${R_1} \leftrightarrow {R_1} + 5{R_{_2}}$we got the matrix given in option B. So, this matrix is an elementary matrix as it obtained by applying only one elementary operation.
Now in the option C, ${a_{11}} = 0\& {a_{22}} = 0$, so this is not an elementary matrix. As if we apply elementary operation then the value of ${a_{11}} = k\& {a_{22}} = k$, k can be any constant but not zero.
Now in the option D, ${a_{11}} = 1,{a_{22}} = 1\& {a_{33}} = 1$,so this can be an elementary matrix. But to obtain this matrix we have to apply two elementary operations. So, this is not an elementary matrix. By applying single operation we got,
\[ \Rightarrow A = \left( {\begin{array}{*{20}{c}}
  1&0&0 \\
  0&1&0 \\
  0&0&1
\end{array}} \right) \leftrightarrow \left( {\begin{array}{*{20}{c}}
  1&0&0 \\
  0&1&0 \\
  0&5&1
\end{array}} \right)\], ${R_3} \leftrightarrow {R_3} + 5{R_{_2}}$

So, B is the correct option.

Note: Now if we see options A and C than there are one and two diagonal elements are zero respectively. So, they, not an elementary matrix and this option are cancelled. Now option B and D are left, these options can be checked by applying elementary row operation. If the matrix obtained by applying a single elementary operation then the matrix is the elementary matrix. Option D can be cancelled as it is observed that there are changes in 2 elements and is obtained by applying more than matrix operation.