Question

If A is invertible then adj(A) is invertible

true or false? can you explain

Answer 1

@thomaster

Answer 2

true. explain
A is invertible --> \(A^{-1}\) is invertible, too.
\(A^{-1}= \dfrac{1}{det A}*adj (A)\) since A is invertible det (A) \(\neq\)0 --> \(\dfrac{1}{det (A)}\) is a number and \(\dfrac{1}{det(A)}*adj (A) = adj [(number*(A))] \)
therefore \(A^{-1}= another matrix\) \(A^{-1}\) is invertible \(\rightarrow\) "anothermatrix" is invertible or adj (A) is invertible

Answer 3

thanks!

Answer 4

can you help me with one more?

Answer 5

If A has a zero entry on the diagonal then A is not invertible.

Answer 6

just give you a counterexample
\[\left[\begin{matrix}0&1\\-4&0\end{matrix}\right]\] which is 0 in diagonal line
\[A^{-1} = \left[\begin{matrix}0&-0.25\\1&0\end{matrix}\right]\]
therefore, the zero in main diagonal line doesn't decide whether the matrix is invertible or not.

Answer 7

thanks dude u saved me!