Question

Question: (a) Prove that the set Sof matrices of the form $\left(\begin{array}{cc}a& b\\ 0& c\end{array}\right)$with $\mathit{a}\mathbf{,}\mathit{b}\mathbf{,}\mathit{c}\mathbf{\in }\mathbf{ℝ}$is asubring of $\mathit{M}\left(ℝ\right)$.

Short answer

It is proved that the set of matrices with$a,b,c\in R$ is a subring.

Step-by-step solution

Determine Theorem 3.2

Consider that $\left[\begin{array}{cc}a& b\\ 0& c\end{array}\right],\left[\begin{array}{cc}d& e\\ 0& f\end{array}\right]\in T$then,

$\left[\begin{array}{cc}a& b\\ 0& c\end{array}\right]+\left[\begin{array}{cc}d& e\\ 0& f\end{array}\right]=\left[\begin{array}{cc}a+d& b+e\\ 0& c+f\end{array}\right]$

Thus, S is closed under addition.

Consider that,$\left[\begin{array}{cc}a& b\\ 0& c\end{array}\right],\left[\begin{array}{cc}d& e\\ 0& f\end{array}\right]\in S$then,

$\left[\begin{array}{cc}a& b\\ 0& c\end{array}\right]\left[\begin{array}{cc}d& e\\ 0& f\end{array}\right]=\left[\begin{array}{cc}ad& ae+bf\\ 0& ce\end{array}\right]$

Thus, S is closed under multiplication.

Hence, the above addition and multiplication satisfies all the conditions of theorem 3.2.

Determine a,b,c∈ℝis subring of M(ℝ)

Note that, $\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\in S$

for$\left[\begin{array}{cc}a& b\\ 0& c\end{array}\right]\in S$then it would be,

$\left[\begin{array}{cc}-a& -b\\ 0& -c\end{array}\right]\in S$and $\left[\begin{array}{cc}a& b\\ 0& c\end{array}\right]+\left[\begin{array}{cc}-a& -b\\ 0& -c\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

Therefore, the set of matrices with$a,b,c\in R$ is a subring.