deriving the determinant and inverse of a generic 2x2 matrix | Mathematics | Page 1

vital cedar Mar 21, 2026, 8:53 AM

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can someone help me through the process of deriving the formula of the determinant and hence inverse of any 2x2 matrix

pallid stirrupBOT Mar 21, 2026, 8:53 AM

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pallid stirrupBOT Mar 21, 2026, 8:53 AM

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pallid stirrup

vital cedar Mar 21, 2026, 8:56 AM

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i dont understand

torpid harness Mar 21, 2026, 11:38 AM

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vital cedar

They solved the linear system

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what are you confused about exactly ?

vital cedar Mar 21, 2026, 9:28 PM

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seems simple but how did they solve the system of equations for x, u, v and y

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also what is the importance of the determinant

vital cedar Mar 21, 2026, 9:29 PM

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vital cedar also what is the importance of the determinant

other than telling us if theres a possible inverse for a matrix

glacial bay Mar 22, 2026, 12:06 AM

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vital cedar other than telling us if theres a possible inverse for a matrix

Each term of the matrix has a factor 1/(ad-bc) which is 1/determinant of the original matrix

vital cedar Mar 22, 2026, 12:16 AM

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isnt 1/(ad-bc) times the original matrix with a and d switched and c and b minused

glacial bay Mar 22, 2026, 12:19 AM

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vital cedar isnt 1/(ad-bc) times the original matrix with a and d switched and c and b minus...

Yes

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That's the inverse

vital cedar Mar 22, 2026, 6:50 AM

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what does the determinant refer to visually?

quiet kayak Mar 22, 2026, 9:08 AM

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vital cedar what does the determinant refer to visually?

determinants can be visualized as the factor by which the unit area is scaled when a linear transformation takes place

vital cedar Mar 22, 2026, 10:01 AM

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ok but then am i correct in assuming there is a factor which represents the shaw of it

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like how far it turns into a parallelogram

vital cedar Mar 22, 2026, 10:12 AM

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quiet kayak determinants can be visualized as the factor by which the unit area is scaled wh...

thanks for this though it really helped my understanding of the topic

pallid stirrupBOT Mar 23, 2026, 11:00 AM

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@vital cedar

:HelpIcon:| Help Reminder

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pallid stirrupBOT Mar 23, 2026, 2:32 PM

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@quiet kayak @glacial bay @torpid harness The user still needs help with this help request.

quiet kayak Mar 23, 2026, 3:24 PM

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vital cedar ok but then am i correct in assuming there is a factor which represents the shaw...

I mean you could see the basis from columns and find out the angles ig

vital cedar Mar 24, 2026, 5:26 AM

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alright so its not a set expression or

paper grail Mar 25, 2026, 6:44 PM

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vital cedar can someone help me through the process of deriving the formula of the determina...

Use a system of equations

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Let A be a 2x2 matrix

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A^(-1) is the matrix that satisifies

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A*A^(-1) = I_2

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Where I_2 is the identity matrix of order 2

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So you have

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$$A = \begin{pmatrix}
a & b \
c & d
\end{pmatrix}\
A^{-1}=\begin{pmatrix}
a' & b' \
c' & d'\end{pmatrix}\
\text{Such that:}\
A \cdot A^{-1} = I_2$$

dense peakBOT Mar 25, 2026, 6:52 PM

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._kamikaze._

paper grail Mar 25, 2026, 6:53 PM

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We now multiple the two matrices

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$$A\cdot A^{-1} = \begin{pmatrix}
aa' +bc' & ab' + bd'\
ca' + dc' & cb'+dd'\end{pmatrix}\
= \begin{pmatrix}
1 & 0\
0 & 1
\end{pmatrix}

dense peakBOT Mar 25, 2026, 6:58 PM

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._kamikaze._
Compile Error! Click the errors reaction for more information.
(You may edit your message to recompile.)

paper grail Mar 25, 2026, 6:58 PM

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Now we solve the system

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$$\begin{cases}
aa' + bc' = 1\
ab' + bd' = 0\
ca' + dc' = 0\
cb' + dd' = 1
\end{cases}$$

dense peakBOT Mar 25, 2026, 6:59 PM

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._kamikaze._

paper grail Mar 25, 2026, 7:01 PM

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We are looking for a', c', b' and d'

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$$
\begin{cases}
aa' + bc' = 1\
ca' + dc' = 0
\end{cases}
\ and \
\begin{cases}
ab' + bd' = 0\
cb' + dd' = 1
\end{cases}
$$

dense peakBOT Mar 25, 2026, 7:03 PM

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._kamikaze._

paper grail Mar 25, 2026, 7:12 PM

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We will work with elimination

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$$
\begin{cases}
caa' + cbc' = c\
aca' + adc' = 0
\end{cases}
$$

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we multiply first line by c and second one by a

dense peakBOT Mar 25, 2026, 7:14 PM

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._kamikaze._

paper grail Mar 25, 2026, 7:14 PM

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We substract (2) from (1)

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(n) corresponds to the nth line

dense peakBOT Mar 25, 2026, 7:15 PM

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._kamikaze._

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._kamikaze._

paper grail Mar 25, 2026, 7:15 PM

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$$caa' + cbc' - caa' -adc' = c$$

dense peakBOT Mar 25, 2026, 7:15 PM

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._kamikaze._

paper grail Mar 25, 2026, 7:15 PM

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Which turns into

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$$ cbc' - adc' = c$$

dense peakBOT Mar 25, 2026, 7:16 PM

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._kamikaze._

paper grail Mar 25, 2026, 7:16 PM

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We factor out the c'

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$$c'(cb-ad) = c$$

dense peakBOT Mar 25, 2026, 7:16 PM

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._kamikaze._

paper grail Mar 25, 2026, 7:16 PM

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thus $c' = \dfrac{c}{cb-ad}$

dense peakBOT Mar 25, 2026, 7:16 PM

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._kamikaze._

paper grail Mar 25, 2026, 7:16 PM

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We do this for all 4 variables

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To find the following inverse matrix

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$$\begin{pmatrix}
\dfrac{d}{ad-bc} & \dfrac{-b}{ad-bc}\
\dfrac{-c}{ad-bc} & \dfrac{a}{ad-bc}
\end{pmatrix}
$$

dense peakBOT Mar 25, 2026, 7:17 PM

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._kamikaze._

paper grail Mar 25, 2026, 7:18 PM

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We notice that the repeating ad-bc is the det(A)

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$$\begin{pmatrix}
\dfrac{d}{\det(A)} & \dfrac{-b}{\det(A)}\
\dfrac{-c}{\det(A)} & \dfrac{a}{\det(A)}
\end{pmatrix}
$$

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And we can now factor it out

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to find

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$$A^{-1} = \dfrac{1}{\det(A)}\begin{pmatrix}
d & -b\
-c} & a
\end{pmatrix}$$

dense peakBOT Mar 25, 2026, 7:19 PM

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._kamikaze._
Compile Error! Click the errors reaction for more information.
(You may edit your message to recompile.)

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._kamikaze._

paper grail Mar 25, 2026, 7:20 PM

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Nice

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i think the bot bugged out

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Nvm it didnt

twin snow Mar 25, 2026, 8:39 PM

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paper grail Use a system of equations

#bot-cmd-latex to spam LaTeX, also observe they already got help.

paper grail Mar 25, 2026, 8:46 PM

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twin snow <#696291655082442762> to spam LaTeX, also observe they already got help.

How am I spamming?

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I don't see how helping twice is a bad thing considering the thread is still open

twin snow Mar 25, 2026, 9:53 PM

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paper grail How am I spamming?

You did not work with the student, you just left them a bunch of LaTeX entries to read on their own, and have to sift through them and the actual sentences.

As for the channel still being open... they just never closed in. But again, reading the channel would clearly tell you they got help and were satisfied with the help given.

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Also per the rules, one helper per channel unless the channel has been ghosted by the original helper (and OP is clearly still present.)

paper grail Mar 25, 2026, 10:32 PM

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Ok

vital cedar Mar 26, 2026, 2:46 AM

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I found it helpful how they explained how they solved it

#deriving the determinant and inverse of a generic 2x2 matrix