Let $n$ be an odd positive integer.
Determine whether there exists an $n \times n$ real matrix $A$ such that
\[A^2+I=O,\]
where $I$ is the $n \times n$ identity matrix and $O$ is the $n \times n$ zero matrix.

If such a matrix $A$ exists, find an example. If not, prove that there is no such $A$.

The key technique to solve this problem very easily is determinant.
Recall the following properties of the determinant.
Let $A, B$ be $n\times n$ matrices and $c$ be a scalar Then we have

$\det(AB)=\det(A)\det(B)$

$\det(cA)=c^n\det(A)$.

Solution.

When $n$ is odd.

When $n$ is odd, we prove that there is no $n \times n$ real matrix $A$ such that $A^2+I=O$.
Seeking a contradiction, assume that we have $A$ such that $A^2+I=O$.
Since we have $A^2=-I$, we have
\[\det(A^2)=\det(-I).\]

Using the properties of determinant, we obtain
\[\det(A)^2=(-1)^n\det(I)=-1\]
because $n$ is odd and $\det(I)=1$.

Since $A$ is a real matrix, the determinant of $A$ is also real.
Thus, $\det(A)^2=-1$ is impossible. Hence there is no such $A$.

When $n$ is even.

On the other hand, if $n$ is even there is $A$ such that $A^2+I=O$.
For example, consider
\[A=\begin{bmatrix}
0 & -1\\
1& 0
\end{bmatrix}.\]
Then a direct computation shows that $A^2=-I$, hence $A^2+I=O$.

Comment.

Recall that the imaginary number $i$ is the number whose square is $-1$.
Similarly, we found above that the square of the matrix $A=\begin{bmatrix}
0 & -1\\
1& 0
\end{bmatrix}$ is $-I$.

But when $n$ is odd, there is no such matrix $A$ as we showed.

10 Examples of Subsets that Are Not Subspaces of Vector Spaces
Each of the following sets are not a subspace of the specified vector space. For each set, give a reason why it is not a subspace.
(1) \[S_1=\left \{\, \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} \in \R^3 \quad \middle | \quad x_1\geq 0 \,\right \}\]
in […]

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 […]

Use Cramer’s Rule to Solve a $2\times 2$ System of Linear Equations
Use Cramer's rule to solve the system of linear equations
\begin{align*}
3x_1-2x_2&=5\\
7x_1+4x_2&=-1.
\end{align*}
Solution.
Let
\[A=[A_1, A_2]=\begin{bmatrix}
3 & -2\\
7& 4
\end{bmatrix},\]
be the coefficient matrix of the system, where $A_1, A_2$ […]

If 2 by 2 Matrices Satisfy $A=AB-BA$, then $A^2$ is Zero Matrix
Let $A, B$ be complex $2\times 2$ matrices satisfying the relation
\[A=AB-BA.\]
Prove that $A^2=O$, where $O$ is the $2\times 2$ zero matrix.
Hint.
Find the trace of $A$.
Use the Cayley-Hamilton theorem
Proof.
We first calculate the […]

Linear Transformation $T(X)=AX-XA$ and Determinant of Matrix Representation
Let $V$ be the vector space of all $n\times n$ real matrices.
Let us fix a matrix $A\in V$.
Define a map $T: V\to V$ by
\[ T(X)=AX-XA\]
for each $X\in V$.
(a) Prove that $T:V\to V$ is a linear transformation.
(b) Let $B$ be a basis of $V$. Let $P$ be the matrix […]

Eigenvalues of a Matrix and Its Squared Matrix
Let $A$ be an $n \times n$ matrix. Suppose that the matrix $A^2$ has a real eigenvalue $\lambda>0$. Then show that either $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of the matrix $A$.
Hint.
Use the following fact: a scalar $\lambda$ is an eigenvalue of a […]

Special Linear Group is a Normal Subgroup of General Linear Group
Let $G=\GL(n, \R)$ be the general linear group of degree $n$, that is, the group of all $n\times n$ invertible matrices.
Consider the subset of $G$ defined by
\[\SL(n, \R)=\{X\in \GL(n,\R) \mid \det(X)=1\}.\]
Prove that $\SL(n, \R)$ is a subgroup of $G$. Furthermore, prove that […]