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Q38E

Expert-verified
Found in: Page 307

Linear Algebra With Applications

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 ${\mathbf{adj}}\left(A\right){\mathbf{,}}{\mathbf{adj}}\left(B\right){\mathbf{,}}{\mathbf{}}{\mathbf{and}}{\mathbf{}}{\mathbit{a}}{\mathbit{d}}{\mathbit{j}}\left(AB\right)$?

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

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

See the step by step solution

Step 1: Matrix Definition.

Matrix is a set of numbers arranged in rows and columns so as to form a rectangular array.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be an “m by n” matrix, written “.mxn”

Step 2: To find the relationship of  and

To find the relationship,

It applies,

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