Question

Find the $B$ matrix for the transformation$\mathrm{x}\mapsto A\mathrm{x}$, when $B=\left\{b_1,b_2,b_3\right\}$.

$A=\left(\begin{array}{rlrlr}-14& & 4& & -14\\ -33& & 9& & -31\\ 11& & -4& & 11\end{array}\right)$, $b_1=\left(\begin{array}{r}-1\\ -2\\ 1\end{array}\right)$, $b_2=\left(\begin{array}{r}-1\\ -1\\ 1\end{array}\right)$, $b_3=\left(\begin{array}{r}-1\\ -2\\ 0\end{array}\right)$

Short answer

The B matrix of the transformation $\mathrm{x}\mapsto A\mathrm{x}$is $D=P^{-1}AP$, which is $\left(\begin{array}{rlrlr}8& & 3& & -6\\ 0& & 1& & 3\\ 0& & 0& & -3\end{array}\right)$

Step-by-step solution

Find P matrix

Let $P$ is an invertible matrix and $D$ is the diagonal matrix, such that the matrix A can be written as follows:

$A=PD{P}^{-1}$

Where the columns of the $P$ matrix are the same as that of basis $B$. So, the matrix $P$ can be written as:

$\begin{array}{rl}P& =\left({\mathrm{b}}_1{\mathrm{b}}_2{\mathrm{b}}_3\right)\\ & =\left(\begin{array}{rlrlr}-1& & -1& & -1\\ -2& & -1& & -2\\ 1& & 1& & 0\end{array}\right)\end{array}$

Find D matrix

As $A=PD{P}^{-1}$ then the diagonal matrix $D$ can be obtained as $D=P^{-1}AP$. To find $D=P^{-1}AP$, first, find $P^{-1}$using the cofactors and determinant method.

$\begin{array}{r}P=\left(\begin{array}{rlrlr}-1& & -1& & -1\\ -2& & -1& & -2\\ 1& & 1& & 0\end{array}\right)\\ P^{-1}=\left(\begin{array}{rlrlr}2& & -1& & 1\\ -2& & 1& & 0\\ -1& & 0& & -1\end{array}\right)\end{array}$

Now find the matrix $D$ as follows:

$\begin{array}{rl}D& =P^{-1}AP\\ & =\left(\begin{array}{rlrlr}2& & -1& & 1\\ -2& & 1& & 0\\ -1& & 0& & -1\end{array}\right)\left(\begin{array}{rlrlr}-14& & 4& & -14\\ -33& & 9& & -31\\ 11& & -4& & 11\end{array}\right)\left(\begin{array}{rlrlr}-1& & -1& & -1\\ -2& & -1& & -2\\ 1& & 1& & 0\end{array}\right)\\ & =\left(\begin{array}{rlrlr}8& & 3& & -6\\ 0& & 1& & 3\\ 0& & 0& & -3\end{array}\right)\end{array}$