# Which of the following matrices is elementary matrix? a) begin{bmatrix}0 & 3 1 & 0 end{bmatrix} b) begin{bmatrix}2 & 0 0 & 1 end{bmatrix} c) begin{bmatrix}1 & 0 0 & 1 end{bmatrix} d) begin{bmatrix}2 & 0 0 & 2 end{bmatrix}

Which of the following matrices is elementary matrix?
a) $\left[\begin{array}{cc}0& 3\\ 1& 0\end{array}\right]$
b) $\left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right]$
c) $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
d) $\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$
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Step 1
Since your question has multiple subparts we have solved the first three subparts for you.
Identify the elementary matrices
Elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation.
a) $\left[\begin{array}{cc}0& 3\\ 1& 0\end{array}\right]$
Apply minimum elemantary row operation on identity matrix to form this matrix.
$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
${R}_{1}↔{R}_{2}$
$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$
${R}_{1}↔3{R}_{1}$
$\left[\begin{array}{cc}0& 3\\ 1& 0\end{array}\right]$
Since the matrix is formed by applying two elementary row operations therefore it is not an elementary matrix.
Step 2
Similarly check other matrices
b) $\left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right]$
$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
${R}_{1}↔{R}_{1}$
$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
Since $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ differs the identity matrix by one elementary operation hence it is an elementary matrix.
Step 3 Similarly check other matrices
c) $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right]$
${R}_{1}↔2{R}_{1}$
$\left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right]$
Since $\left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right]$ differs the identity matrix by one elementary operation hence it is an elementary matrix.
Jeffrey Jordon