StudySmarter AI is coming soon!

- :00Days
- :00Hours
- :00Mins
- 00Seconds

A new era for learning is coming soonSign up for free

Suggested languages for you:

Americas

Europe

Q45E

Expert-verifiedFound in: Page 308

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**Find all ${\mathbf{2}}{\mathbf{\times}}{\mathbf{2}}$**** matrices A such that ${\mathbf{adj}}{\left(A\right)}{\mathbf{=}}{{\mathit{A}}}^{{\mathbf{T}}}$**

Therefore, the $\mathrm{adj}\left(A\right)={A}^{T}$ is given by,

$A=\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]$.

Let,

$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

Then, we have

${A}^{T}=\left[\begin{array}{cc}a& c\\ b& d\end{array}\right]$ ,

and

$\mathrm{adj}A=\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

For ${A}^{T}=\mathrm{adj}A$, we need $a=d$ and $b=-c$.

Thus,

$A=\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]$ .

94% of StudySmarter users get better grades.

Sign up for free