solve the matrix equation.\n17. $\begin{bmatrix}10&1\\5&xend{bmatrix}-2\begin{bmatrix}9&1\\y&5end{bmatrix}=\b…

Explanation:

Step1: Expand the left - hand side of the first equation

First, calculate (2\begin{bmatrix}9&1\y&5\end{bmatrix}=\begin{bmatrix}2\times9&2\times1\2\times y&2\times5\end{bmatrix}=\begin{bmatrix}18&2\2y&10\end{bmatrix}). Then (\begin{bmatrix}10&1\5&x\end{bmatrix}-\begin{bmatrix}18&2\2y&10\end{bmatrix}=\begin{bmatrix}10 - 18&1 - 2\5-2y&x - 10\end{bmatrix}=\begin{bmatrix}-8&-1\5 - 2y&x - 10\end{bmatrix}). Since (\begin{bmatrix}-8&-1\5 - 2y&x - 10\end{bmatrix}=\begin{bmatrix}-8&-1\11&-20\end{bmatrix}), we can set up two equations: For the bottom - left elements: (5-2y = 11), and for the bottom - right elements: (x - 10=-20).

Step2: Solve for (x)

From (x - 10=-20), add 10 to both sides of the equation: (x=-20 + 10=-10).

Step3: Solve for (y)

From (5-2y = 11), first subtract 5 from both sides: (-2y=11 - 5 = 6). Then divide both sides by (-2), so (y=\frac{6}{-2}=-3).

Step4: Solve the second equation for (X)

For the equation (\begin{bmatrix}1&-2\8&5\end{bmatrix}-3X=\begin{bmatrix}-11&4\8&-16\end{bmatrix}), first isolate (3X): (3X=\begin{bmatrix}1&-2\8&5\end{bmatrix}-\begin{bmatrix}-11&4\8&-16\end{bmatrix}). Calculate the right - hand side: (\begin{bmatrix}1-(-11)&-2 - 4\8 - 8&5-(-16)\end{bmatrix}=\begin{bmatrix}12&-6\0&21\end{bmatrix}). Then (X=\frac{1}{3}\begin{bmatrix}12&-6\0&21\end{bmatrix}=\begin{bmatrix}\frac{12}{3}&\frac{-6}{3}\\frac{0}{3}&\frac{21}{3}\end{bmatrix}=\begin{bmatrix}4&-2\0&7\end{bmatrix}).

Answer:

For the first equation, (x=-10,y = - 3). For the second equation, (X=\begin{bmatrix}4&-2\0&7\end{bmatrix})