Question

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

Short answer

It is proved that 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, T is closed under addition.

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

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

Thus,T is closed under multiplication.

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

Determine a,b,c∈ℝis subring of M(ℝ)

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

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

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

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