Let M be the vector space of 2 times 2 real-valued matrices.M=begin{bmatrix}a & b c & d end{bmatrix}and define M^{#}=begin{bmatrix}d & b c & a end{bmatrix}

Jaya Legge

Jaya Legge

Answered question

2020-11-30

Let M be the vector space of 2×2 real-valued matrices.
M=[abcd]
and define M#=[dbca] Characterize the matrices M such that M#=M1

Answer & Explanation

FieniChoonin

FieniChoonin

Skilled2020-12-01Added 102 answers

Step 1
Let M be the vector space of 2×2 real-valued matrices.
M=[abcd]
and define M#=[dbca]
To characterize matrix M such that M#=M1
Therefore for findiny M1 we want det(M)0
Therefore det(M)=adbc0
If det(M)0
then M1=1adbc[dbca]
So, M#=M1
[dbca]=1(adbc)[dbca]
[(adbc)d(adbc)b(adbc)c(adbc)a]=[dbca]
Case (1)
By comparing if either a=0 pr d=0 of both then abbc=bc0(as adbc0)
So, b and c both non zero
b=b(adbc)abbc=1
Step 2
M#=M1 and if either a=0 or d=0 of both then ad-bc=-1
Case ( 2 )
If a,d0 then
a=(adbc)aadbc=1
So, b=b and c=c
2b=0 and 2c=0
b=0c=0
So, M=[a00d] with ad=1

Jeffrey Jordon

Jeffrey Jordon

Expert2022-01-24Added 2605 answers

Answer is given below (on video)

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