# Find a family of the matrices that is similar to the matrix Q=begin{bmatrix}p & -2 4 & -3 end{bmatrix}

Find a family of the matrices that is similar to the matrix $Q=\left[\begin{array}{cc}p& -2\\ 4& -3\end{array}\right]$
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Nicole Conner
Step 1
The given matrix Q is :
$Q=\left[\begin{array}{cc}p& -2\\ 4& -3\end{array}\right]$
Consider a non singular matrix :
$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$
The similar matrix for $Q=AQ{A}^{-1}=B$
Therefore, B is the similar matrix to Q.
$B=AQ{A}^{-1}$
$Q={A}^{-1}BA$
Step 2
Now value for A^{-1} ,
${A}^{-1}=\frac{adj\left(A\right)}{|A|}$
$=\frac{1}{-2}\left[\begin{array}{cc}4& -2\\ -3& 1\end{array}\right]$
$=\left[\begin{array}{cc}-2& 1\\ \frac{3}{2}& -\frac{1}{2}\end{array}\right]$
Step 3
Therefore, the value for B,
$B=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\left[\begin{array}{cc}p& -2\\ 4& -3\end{array}\right]\left[\begin{array}{cc}-2& 1\\ \frac{3}{2}& -\frac{1}{2}\end{array}\right]$
$=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\left[\begin{array}{cc}-2p-3& p+1\\ -\frac{25}{2}& \frac{11}{2}\end{array}\right]$
$=\left[\begin{array}{cc}-2p-28& p+12\\ -6p-59& p+23\end{array}\right]$
This matrix B is the family of singular matrices.
Jeffrey Jordon