4. In each of the following, find a system of equations and find [tex]$x$[/tex] and [tex]$y$[/tex].
a. [tex]$\left(\begin{array}{cc}3 & 0 \\ 0 & -2\end{array}\right)\binom{x}{y}=\binom{12}{8}$[/tex]
b. [tex]$\left(\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right)\binom{x}{y}=\binom{8}{1}$[/tex]
c. [tex]$\left(\begin{array}{ll}x & y \\ y & x\end{array}\right)\binom{3}{1}=\binom{5}{-1}$[/tex]
d. [tex]$\left(\begin{array}{cr}x+1 & 3 \\ \frac{1}{2} & -y\end{array}\right)\binom{2}{1}=\binom{11}{0}$[/tex]
5. Given [tex]$\left(\begin{array}{rr}3 & 1 \\ 2 & -1\end{array}\right)\binom{x}{y}=\binom{9}{1}$[/tex] find a system of equations in [tex]$x$[/tex] and [tex]$y$[/tex]. Hence find [tex]$x$[/tex] and [tex]$y$[/tex].
6. If [tex]$A=\left(\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right)$[/tex], find [tex]$p, q$[/tex] such that [tex]$A^2=p A+q I$[/tex]. ([tex]$A^2=A A$[/tex] and [tex]$I=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$[/tex].
Answer (2)
See the below works. ;
The solution involved writing the given matrix equations as systems of linear equations and solving for variables x and y. Each part provided unique equations that led to specific values for x and y. The methods employed matrix multiplication and algebraic manipulation to find the solutions effectively.
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