Scalar Matrix Exercise (with Solution)

Exercise:

\[\] \[ \begin{array}{|c|c|} \hline \textbf{Matrix} & \textbf{Scalar Matrix?} \\ \hline \begin{pmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{pmatrix} & ? \\ \hline \begin{pmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{pmatrix} & ? \\ \hline \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{pmatrix} & ? \\ \hline \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} & ? \\ \hline \begin{pmatrix} 7 & 2 & 0 \\ 0 & 7 & 1 \\ 0 & 0 & 7 \end{pmatrix} & ? \\ \hline \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} & ? \\ \hline \end{array} \]

Solution:

\[\] \[ \begin{array}{|c|c|} \hline \textbf{Matrix} & \textbf{Scalar Matrix?} \\ \hline \begin{pmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{pmatrix} & \text{Yes} \\ \hline \begin{pmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{pmatrix} & \text{Yes} \\ \hline \begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{pmatrix} & \text{No} \\ \hline \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} & \text{Yes} \\ \hline \begin{pmatrix} 7 & 2 & 0 \\ 0 & 7 & 1 \\ 0 & 0 & 7 \end{pmatrix} & \text{No} \\ \hline \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} & \text{Yes} \\ \hline \end{array} \]