Question

If$\mathbf{A}$ is an $\mathbf{}\mathbf{n}\mathbf{\times }\mathbf{n}$ matrix of rank $\mathbf{n}\mathbf{-}\mathbf{1}$, what is the rank of $\mathbf{adj}\mathbf{\left(}\mathbf{A}\mathbf{\right)}$? See Exercises 42 and 43.

Short answer

Therefore, the rank of $\mathrm{adj}(\mathrm{A})$is given by,

$\mathrm{dimim}(\mathrm{adj}\mathrm{A})=1$

Step-by-step solution

Matrix Definition. 

Matrix is a set of numbers arrangedin rows and columns soas to form a rectangulararray.

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 “.$\mathrm{m}\times \mathrm{n}$”

What is the rank of adj(A).

Ifis a non-invertible matrix, then $A(adjA)=0.$.

Therefore, $\mathrm{im}(\mathrm{adj}\mathrm{A})$is a subspace of $\mathrm{kerA}$.

Thus,

$$\begin{gathered} dimim(adjA)\le dimkerA \\ dimim\left(adjA\right)=n-dimimA \\ dimim(adjA)=n-(n-1) \\ dimim(adjA)=1 \end{gathered}$$

and since the rank of any matrix can't be 0.

We conclude $dimim(adjA)=1$.