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Q38E

Expert-verifiedFound in: Page 307

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**For two invertible nxn**** matrices** A **and** B **, what is the relationship between ${\mathbf{adj}}{\left(A\right)}{\mathbf{,}}{\mathbf{adj}}{\left(B\right)}{\mathbf{,}}{\mathbf{}}{\mathbf{and}}{\mathbf{}}{\mathit{a}}{\mathit{d}}{\mathit{j}}{\left(AB\right)}$****?**

Therefore, the relationship of $adj\left(A\right),adj\left(B\right)\mathrm{and}\mathrm{adj}\left(\mathrm{AB}\right)$ is given by,

$\mathrm{adj}\left(\mathrm{AB}\right)=\mathrm{adj}\left(\mathrm{B}\right)\mathrm{adj}\left(\mathrm{A}\right).$

To find the relationship,

It applies,

$\mathrm{adj}\left(\mathrm{AB}\right)=\left(\mathrm{det}(\mathrm{AB}\right){\left(\mathrm{AB}\right)}^{-1}\mathrm{adj}\left(\mathrm{AB}\right)=\left(\mathrm{det}\mathrm{A}\mathrm{det}\mathrm{B}\right){B}^{-1}{A}^{-1}\phantom{\rule{0ex}{0ex}}\mathrm{adj}\left(\mathrm{AB}\right)=\left(\left(\mathrm{det}\mathrm{B}\right){\mathrm{B}}^{-1}\right)\left(\left(\mathrm{det}\mathrm{A}\right){\mathrm{A}}^{-1}\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Therefore}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{adj}\left(\mathrm{AB}\right)=\mathrm{adj}\left(\mathrm{B}\right)\mathrm{adj}\left(\mathrm{A}\right).$

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