If A = [ ll 1 & 3 3 & 4 ] B = [ ll -2 & 1 -3 & 2 ] and A^2-5 B^2=5 C Find the matrix C where C is a 2 by 2 matrix.

A^2=A A=p [ ll 1 & 3 3 & 4 ] [ ll 1 & 3 3 & 4 ] [ cc 1 1+3 3 & 1 3+3 4 3 1+4 3 & 3 3+34 40 ] = [ cc 10 & 15 15 & 25 ] l B^2=B B= [ ll -2 & 1 -3 & 2 ] [ ll -2 & 1 -3 & 2 ] = [ ll -2 -2+1 -3 & -2 1+1 2 -3 -2+2 -3 & -3 1+2 2 ] = [ ll 1 & 0 0 & 1 ] Given: A^2-5 B^2=5 C l [ ll 10 & 15 15 & 25 ]-5 [ ll 1 & 0 0 & 1 ]=5 C [ ll 10 & 15 15 & 25 ]- [ ll 5 & 0 0 & 5 ]=5 C [ cc 5 & 15 15 & 20 ]=5 C 5 [ ll 1 & 3 3 & 4 ]=5 C C= [ ll 1 & 3 3 & 4 ]

If A = [ ll 1 & 3 3 & 4 ] B = [ ll -2 & 1 -3 & 2 ] and A^2-5 B^2=…

Download App

Question

If $A =\left[\begin{array}{ll}1 & 3 \\ 3 & 4\end{array}\right] B =\left[\begin{array}{ll}-2 & 1 \\ -3 & 2\end{array}\right]$ and $A^2-5 B^2=5 C$ Find the matrix $C$ where $C$ is a $2$ by $2$ matrix.

✓

Answer

$A^2=A \times A=p\left[\begin{array}{ll}1 & 3 \\ 3 & 4\end{array}\right]\left[\begin{array}{ll}1 & 3 \\ 3 & 4\end{array}\right]$
$\left[\begin{array}{cc}1 \times 1+3 \times 3 & 1 \times 3+3 \times 4 \\ 3 \times 1+4 \times 3 & 3 \times 3+34 \times 40\end{array}\right]$
$=\left[\begin{array}{cc}10 & 15 \\ 15 & 25\end{array}\right]$
$\begin{array}{l}B^2=B \times B=\left[\begin{array}{ll}-2 & 1 \\ -3 & 2\end{array}\right]\left[\begin{array}{ll}-2 & 1 \\ -3 & 2\end{array}\right]\end{array}$
$=\left[\begin{array}{ll}-2 \times-2+1 \times-3 & -2 \times 1+1 \times 2 \\ -3 \times-2+2 \times-3 & -3 \times 1+2 \times 2\end{array}\right]$
$=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
Given: $A^2-5 B^2=5 C$
$\begin{array}{l}\Rightarrow\left[\begin{array}{ll}10 & 15 \\ 15 & 25\end{array}\right]-5\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=5 C\end{array} $
$\Rightarrow\left[\begin{array}{ll}10 & 15 \\ 15 & 25\end{array}\right]-\left[\begin{array}{ll}5 & 0 \\ 0 & 5\end{array}\right]=5 C$
$\Rightarrow\left[\begin{array}{cc}5 & 15 \\ 15 & 20\end{array}\right]=5 C $
$ \Rightarrow 5\left[\begin{array}{ll}1 & 3 \\ 3 & 4\end{array}\right]=5 C$
$\Rightarrow C=\left[\begin{array}{ll}1 & 3 \\ 3 & 4\end{array}\right]$