Question

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

\(A = \left( {\begin{aligned}{}{ - 7}&{ - 48}&{ - 16}\\1&{14}&6\\{ - 3}&{ - 45}&{ - 19}\end{aligned}} \right)\),

\({b_1} = \left( {\begin{aligned}{}{ - 3}\\1\\{ - 3}\end{aligned}} \right)\), \({b_2} = \left( {\begin{aligned}{}{ - 2}\\1\\{ - 3}\end{aligned}} \right)\), \({b_3} = \left( {\begin{aligned}{}{\,\,\,3}\\{ - 1}\\{\,\,\,0}\end{aligned}} \right)\)

Short answer

The B matrix of the transformation \({\rm{x}} \mapsto A{\rm{x}}\)is \(D = {P^{ - 1}}AP\), which is \(\left( {\begin{aligned}{}{ - 7}&{}&{ - 2}&{}&{ - 6}\\0&{}&{ - 4}&{}&{ - 6}\\0&{}&0&{}&{ - 1}\end{aligned}} \right)\)

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:

\(\begin{aligned}{c}P &= \left( {{{\rm{b}}_1}\,\,{{\rm{b}}_2}\,\,{{\rm{b}}_3}} \right)\\ &= \left( {\begin{aligned}{}{ - 3}&{}&{ - 2}&{}&3\\1&{}&1&{}&{ - 1}\\{ - 3}&{}&{ - 3}&{}&0\end{aligned}} \right)\end{aligned}\)

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.

\(\begin{aligned}{c}P &= \left( {\begin{aligned}{}{ - 3}&{}&{ - 2}&{}&3\\1&{}&1&{}&{ - 1}\\{ - 3}&{}&{ - 3}&{}&0\end{aligned}} \right)\\{P^{ - 1}} &= \left( {\begin{aligned}{}{ - 1}&{}&{ - 3}&{}&{ - 1/3}\\{\,\,1}&{}&{\,\,3}&{}&{\,0}\\{\,\,0}&{}&{ - 1}&{}&{ - 1/3}\end{aligned}} \right)\end{aligned}\)

Now find the matrix \(D\) as follows:

\(\begin{aligned}{c}D &= {P^{ - 1}}AP\\ &= \left( {\begin{aligned}{}{ - 1}&{}&{ - 3}&{}&{ - 1/3}\\{\,\,1}&{}&{\,\,3}&{}&{\,0}\\{\,\,0}&{}&{ - 1}&{}&{ - 1/3}\end{aligned}} \right)\left( {\begin{aligned}{}{ - 7}&{}&{ - 48}&{}&{ - 16}\\1&{}&{14}&{}&6\\{ - 3}&{}&{ - 45}&{}&{ - 19}\end{aligned}} \right)\left( {\begin{aligned}{}{ - 3}&{}&{ - 2}&{}&3\\1&{}&1&{}&{ - 1}\\{ - 3}&{}&{ - 3}&{}&0\end{aligned}} \right)\\ &= \left( {\begin{aligned}{}{ - 7}&{}&{ - 2}&{}&{ - 6}\\0&{}&{ - 4}&{}&{ - 6}\\0&{}&0&{}&{ - 1}\end{aligned}} \right)\end{aligned}\)

The B matrix of the transformation \({\rm{x}} \mapsto A{\rm{x}}\) is \(D = {P^{ - 1}}AP\), which is \(\left( {\begin{aligned}{}{ - 7}&{}&{ - 2}&{}&{ - 6}\\0&{}&{ - 4}&{}&{ - 6}\\0&{}&0&{}&{ - 1}\end{aligned}} \right)\).