Invertible Matrix Satisfying a Quadratic Polynomial

Problem 266

Let $A$ be an $n \times n$ matrix satisfying
\[A^2+c_1A+c_0I=O,\]
where $c_0, c_1$ are scalars, $I$ is the $n\times n$ identity matrix, and $O$ is the $n\times n$ zero matrix.

Prove that if $c_0\neq 0$, then the matrix $A$ is invertible (nonsingular).
How about the converse? Namely, is it true that if $c_0=0$, then the matrix $A$ is not invertible?

Suppose first that $c_0\neq 0$.
Then we have
\begin{align*}
A^2+c_1A=-c_0I\\
\Leftrightarrow A(A+c_1I)=-c_0 I\\
\Leftrightarrow A\left(\frac{-1}{c_0}(A+c_1I) \right)=I.
\end{align*}
It is in the last step that we needed to assume $c_0\neq0$.

Thus, if we put
\[B=\frac{-1}{c_0}(A+c_1I),\]
then we have proved that
\[AB=I.\]

Similarly, one can check that $BA=A$. Hence $B$ is the inverse matrix of $A$.
Namely,
\[A^{-1}=\frac{-1}{c_0}(A+c_1I).\]
This proves that when $c_0\neq 0$ the matrix $A$ is invertible.

Is it true that if $c_0=0$, then the matrix $A$ is not invertible?

Next, let us consider the case $c_0=0$.
We claim that the matrix $A$ can be invertible even $c_0=0$.

For example, if $A=I$, then $A$ satisfies
\[A^2-A=O.\]
(Thus, $c_1=-1$ and $c_0=0$.)

Since the identity matrix is invertible, the condition $c_0=0$ does not force the matrix $A$ to be non-invertible.

Find a Nonsingular Matrix Satisfying Some Relation
Determine whether there exists a nonsingular matrix $A$ if
\[A^2=AB+2A,\]
where $B$ is the following matrix.
If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$.
(a) \[B=\begin{bmatrix}
-1 & 1 & -1 \\
0 &-1 &0 \\
1 & 2 & […]

Problems and Solutions About Similar Matrices
Let $A, B$, and $C$ be $n \times n$ matrices and $I$ be the $n\times n$ identity matrix.
Prove the following statements.
(a) If $A$ is similar to $B$, then $B$ is similar to $A$.
(b) $A$ is similar to itself.
(c) If $A$ is similar to $B$ and $B$ […]

A Matrix is Invertible If and Only If It is Nonsingular
In this problem, we will show that the concept of non-singularity of a matrix is equivalent to the concept of invertibility.
That is, we will prove that:
A matrix $A$ is nonsingular if and only if $A$ is invertible.
(a) Show that if $A$ is invertible, then $A$ is […]

A Matrix Similar to a Diagonalizable Matrix is Also Diagonalizable
Let $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable.
Definitions/Hint.
Recall the relevant definitions.
Two matrices $A$ and $B$ are similar if there exists a nonsingular (invertible) matrix $S$ such […]

Quiz 4: Inverse Matrix/ Nonsingular Matrix Satisfying a Relation
(a) Find the inverse matrix of
\[A=\begin{bmatrix}
1 & 0 & 1 \\
1 &0 &0 \\
2 & 1 & 1
\end{bmatrix}\]
if it exists. If you think there is no inverse matrix of $A$, then give a reason.
(b) Find a nonsingular $2\times 2$ matrix $A$ such that
\[A^3=A^2B-3A^2,\]
where […]

10 True of False Problems about Nonsingular / Invertible Matrices
10 questions about nonsingular matrices, invertible matrices, and linearly independent vectors.
The quiz is designed to test your understanding of the basic properties of these topics.
You can take the quiz as many times as you like.
The solutions will be given after […]

Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations
For each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A|I]$, where $I$ is the $3\times 3$ identity matrix.
(a) $A=\begin{bmatrix}
1 & 3 & -2 \\
2 &3 &0 \\
[…]