Cabiolab

2021-02-15

Given the two matrices,

(a) Find det A, det B , det(AB) , det(BA) , det(5A) , $det{A}^{T}$ and $det\left({B}^{6}\right)$
(c) Find ${A}^{-1}$ and ${B}^{-1}$ using the adjoint matrices you found in part (b)

odgovoreh

Step 1
We have given the matrices

Step 2
Part(a)
Find det A:
$detA=det\left[\begin{array}{ccc}1& 2& 3\\ 1& 1& 2\\ 0& 1& 2\end{array}\right]=1\cdot det\left[\begin{array}{cc}1& 2\\ 1& 2\end{array}\right]-2\cdot det\left[\begin{array}{cc}1& 2\\ 0& 2\end{array}\right]+3\cdot det\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$
$=1\cdot 0-2\cdot 2+3\cdot 1$
=-1 Find det B:
$detB=det\left[\begin{array}{ccc}1& 1& 1\\ 2& 1& 2\\ 3& 1& 2\end{array}\right]$ $=1\cdot det\left[\begin{array}{cc}1& 2\\ 1& 2\end{array}\right]-1\cdot det\left[\begin{array}{cc}2& 2\\ 3& 2\end{array}\right]+1\cdot det\left[\begin{array}{cc}2& 1\\ 3& 1\end{array}\right]$
$=1\cdot 0-1\cdot \left(-2\right)+1\cdot \left(-1\right)$
=1
Step 3
Find det(AB) and det(BA)
According to determinant properties,
$det\left(AB\right)=detA×detB$
$=-1×1$
=-1
$det\left(BA\right)=detB×detA$
$=1×-1$
=-1
Step 4
Find det(5A)
$det\left(5A\right)={5}^{3}×detA$
$=125×-1$
=-125
Find $det{A}^{T}:$
$det{A}^{T}=detA$
=-1
Find $det\left({B}^{6}\right):$
$det\left({B}^{6}\right)=\left(detB{\right)}^{6}$
$={1}^{6}$
=1
Step 5
Part (b)
$A=\left[\begin{array}{ccc}1& 2& 3\\ 1& 1& 2\\ 0& 1& 2\end{array}\right]$
The cofactors matrix is
$C=\left[\begin{array}{ccc}+det\left[\begin{array}{cc}1& 2\\ 1& 2\end{array}\right]& -det\left[\begin{array}{cc}1& 2\\ 0& 2\end{array}\right]& +det\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\\ -det\left[\begin{array}{cc}2& 3\\ 1& 2\end{array}\right]& +det\left[\begin{array}{cc}1& 3\\ 0& 2\end{array}\right]& -det\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]\\ +det\left[\begin{array}{cc}2& 3\\ 1& 2\end{array}\right]& -det\left[\begin{array}{cc}1& 3\\ 1& 2\end{array}\right]& +det\left[\begin{array}{cc}1& 2\\ 1& 1\end{array}\right]\end{array}\right]$
$C=\left[\begin{array}{ccc}+\left(2-2\right)& -\left(2-0\right)& +\left(1-0\right)\\ -\left(4-3\right)& +\left(2-0\right)& -\left(1-0\right)\\ +\left(<\end{array}$

Jeffrey Jordon