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 2020-11-30 Answered

Let M be the vector space of \(2 \times 2\) real-valued matrices.
\(M=\begin{bmatrix}a & b \\c & d \end{bmatrix}\)
and define \(M^{\#}=\begin{bmatrix}d & b \\c & a \end{bmatrix}\) Characterize the matrices M such that \(M^{\#}=M^{-1}\)

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

FieniChoonin
Answered 2020-12-01 Author has 24011 answers

Step 1
Let M be the vector space of \(2 \times 2\) real-valued matrices.
\(M=\begin{bmatrix}a & b \\c & d \end{bmatrix}\)
and define \(M^{\#}=\begin{bmatrix}d & b \\c & a \end{bmatrix}\)
To characterize matrix M such that \(M^{\#}=M^{-1}\)
Therefore for findiny \(M^{-1}\) we want \(det(M) \neq 0\)
Therefore \(det(M)=ad-bc \neq 0\)
If \(det(M) \neq 0\)
then \(M^{-1}=\frac{1}{ad-bc} \begin{bmatrix}d & -b \\ -c & a \end{bmatrix}\)
So, \(M^{\#}=M^{-1}\)
\(\Rightarrow \begin{bmatrix}d & b \\c & a \end{bmatrix}= \frac{1}{(ad-bc)} \begin{bmatrix}d & -b \\ -c & a \end{bmatrix}\)
\(\Rightarrow \begin{bmatrix}(ad-bc)d & (ad-bc)b \\ (ad-bc)c & (ad-bc)a \end{bmatrix} = \begin{bmatrix}d & -b \\ -c & a \end{bmatrix}\)
Case (1)
By comparing if either a=0 pr d=0 of both then \(ab-bc=-bc \neq 0 (\text{as } ad-bc \neq 0 )\)
So, b and c both non zero
\(\Rightarrow -b=b(ad-bc) \neq ab-bc=-1\)
Step 2
\(M^{\#}=M^{-1}\) and if either a=0 or d=0 of both then ad-bc=-1
Case ( 2 )
If \(a,d \neq 0\) then
\(a=(ad-bc)a \Rightarrow ad-bc=1\)
So, \(b=-b \text{ and } c=-c\)
\(\Rightarrow 2b=0\ and\ 2c=0\)
\(\Rightarrow b=0 \Rightarrow c=0\)
So, \(M=\begin{bmatrix}a & 0 \\ 0 & d \end{bmatrix}\) with ad=1

Not exactly what you’re looking for?
Ask My Question
15
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2020-12-07

Let \(A=\begin{bmatrix}1 & 2 \\-1 & 1 \end{bmatrix} \text{ and } C=\begin{bmatrix}-1 & 1 \\2 & 1 \end{bmatrix}\)
a)Find elementary matrices \(E_1 \text{ and } E_2\) such that \(C=E_2E_1A\)
b)Show that is no elementary matrix E such that \(C=EA\)

asked 2020-10-20

In this problem, allow \(T_1: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) and \(T_2: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) be linear transformations. Find \(Ker(T_1), Ker(T_2), Ker(T_3)\) of the respective matrices:
\(A=\begin{bmatrix}1 & -1 \\-2 & 0 \end{bmatrix} , B=\begin{bmatrix}1 & 5 \\-2 & 0 \end{bmatrix}\)

asked 2020-12-25

The \(2 \times 2 \) matrices A and B below are related to matrix C by the equation: \(C=3A-2B\). Which of the following is matrix C.
\(A=\begin{bmatrix}3 & 5 \\-2 & 1 \end{bmatrix} B=\begin{bmatrix}-4 & 5 \\2 & 1 \end{bmatrix}\)
\(\begin{bmatrix}-1 & 5 \\2 & 1 \end{bmatrix}\)
\(\begin{bmatrix}-18 & 5 \\10 & 1 \end{bmatrix}\)
\(\begin{bmatrix}18 & -5 \\-10 & -1 \end{bmatrix}\)
\(\begin{bmatrix}1 & -5 \\-2 & -1 \end{bmatrix}\)

asked 2021-03-07

Find the following matrices:
a) \(A + B\).
(b) \(A - B\).
(c) \(-4A\).
\(A=\begin{bmatrix}2 & -10&-2 \\14 & 12&10\\4&-2&2 \end{bmatrix} , B=\begin{bmatrix}6 & 10&-2 \\0 & -12&-4\\-5&2&-2 \end{bmatrix}\)

asked 2021-01-17

Let V be the vector space of real 2 x 2 matrices with inner product
\((A|B) = tr(B^tA)\).
Let U be the subspace of V consisting of the symmetric matrices. Find an orthogonal basis for \(U^\perp\) where \(U^{\perp}\left\{A \in V |(A|B)=0 \forall B \in U \right\}\)

asked 2021-03-09

Use the graphing calculator to solve if possible
\(A=\begin{bmatrix}1 & 0&5 \\1 & -5&7\\0&3&-4 \end{bmatrix}\\ B=\begin{bmatrix}3 & -5&3 \\2&3&1\\4&1&-3\end{bmatrix}\\ C=\begin{bmatrix}5 & 2&3 \\2& -1&0 \end{bmatrix}\\ D=\begin{bmatrix}5 \\-3\\4 \end{bmatrix}\)
Find the value in row 2 column 3 of \(AB-3B\)

asked 2021-02-15

Given the two matrices,
\(A=\begin{bmatrix}1 & 2&3 \\1 & 1&2\\0&1&2 \end{bmatrix} \text{ and } B=\begin{bmatrix}1 & 1&1 \\2 & 1&2\\3&1&2 \end{bmatrix}\)
(a) Find det A, det B , det(AB) , det(BA) , det(5A) , \(det A^T\) and \(det(B^6)\)
(b) Find adj A and adj B
(c) Find \(A^{-1}\) and \(B^{-1}\) using the adjoint matrices you found in part (b)

...