Let W be the vector space of 3 times 3 symmetric matrices , A in W Then , which of the following is true ? a) A^T=1 b) dimW=6 c) A^{-1}=A d) A^{-1}=A^T

Question

Let W be the vector space of

3 \times 3

symmetric matrices ,

A \in W

Then , which of the following is true ?
a)

A^{T} = 1

b)

d i m W = 6

c)

A^{- 1} = A

d)

A^{- 1} = A^{T}

Answer 1

Step 1
Given that W is a vector space of

3 \times 3

symmetric matrices.
i.e

W = {A \in M_{3 \times 3} | A^{T} = A}

Let

A \in W

where

A = [\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}]

Then

A^{T} = A

[\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}] = [\begin{matrix} a & d & g \\ b & e & h \\ c & f & i \end{matrix}]

This gives b=d , g=c and h=f
So,

W = {W = A \in M_{3 \times 3} | A = [\begin{matrix} a & b & c \\ b & e & f \\ c & f & i \end{matrix}], a, b, c, e, f, i \in R}

and

[\begin{matrix} a & b & c \\ b & e & f \\ c & f & i \end{matrix}] = a [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] + b [\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] + c [\begin{matrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix}] + e [\begin{matrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}] + f [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}] + i [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}]

Step 2
Basis of

W = {[\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}], [\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}], [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}]}

Therefore, dim(W)=6.
So, option (b) is correct.

Answer 2

Jeffrey Jordon

Answered 2022-01-24 Author has 2047 answers

Answer is given below (on video)

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