Polynomials with Matrix Values


\begin{example}
A polynomial expression, such as $x^{2}-5x-2$, can be evaluated...
...
\item Apply \textsf{Evaluate} to the polynomial .
\end{itemize}
\end{example}

$\blacktriangleright$ Evaluate

x2 -5x - 2 =  $\left[\vphantom{
\begin{array}{rr}
2 & -2 \\
-4 & 0
\end{array}
}\right.$$\begin{array}{rr}
2 & -2 \\
-4 & 0
\end{array}$$\left.\vphantom{
\begin{array}{rr}
2 & -2 \\
-4 & 0
\end{array}
}\right]$

x2 -5x - 2x0 = $\left[\vphantom{
\begin{array}{cc}
2 & -2 \\
-4 & 0
\end{array}
}\right.$$\begin{array}{cc}
2 & -2 \\
-4 & 0
\end{array}$$\left.\vphantom{
\begin{array}{cc}
2 & -2 \\
-4 & 0
\end{array}
}\right]$

You can also define the function f (x) = x2 - 5x - 2 and apply Evaluate (twice) to get the following.

$\blacktriangleright$ Evaluate, Evaluate

f$\left(\vphantom{ \left[
\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}
\right] }\right.$$\left[\vphantom{
\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}
}\right.$$\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}$$\left.\vphantom{
\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}
}\right]$$\left.\vphantom{ \left[
\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}
\right] }\right)$ = $\left[\vphantom{
\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}
}\right.$$\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}$$\left.\vphantom{
\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}
}\right]^{{2}}_{}$ -5$\left[\vphantom{
\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}
}\right.$$\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}$$\left.\vphantom{
\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}
}\right]$ -2 = $\left[\vphantom{
\begin{array}{cc}
2 & -2 \\
-4 & 0
\end{array}
}\right.$$\begin{array}{cc}
2 & -2 \\
-4 & 0
\end{array}$$\left.\vphantom{
\begin{array}{cc}
2 & -2 \\
-4 & 0
\end{array}
}\right]$

The expression -5$\left[\vphantom{
\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}
}\right.$$\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}$$\left.\vphantom{
\begin{array}{cc}
1 & 2 \\
4 & 3
\end{array}
}\right]$ - 2 is not, strictly speaking, a proper expression. However, when evaluated by in Scientific Notebook, the 2 is interpreted in this context as $\left[\vphantom{
\begin{array}{cc}
2 & 0 \\
0 & 2
\end{array}
}\right.$$\begin{array}{cc}
2 & 0 \\
0 & 2
\end{array}$$\left.\vphantom{
\begin{array}{cc}
2 & 0 \\
0 & 2
\end{array}
}\right]$, or twice the 2×2 identity matrix.