Singular or non-singular matrices Which of the following matrices are non-singular?


*

*$I + A$ where $A$ not equal to $0$ is a skew-symmetric real $n\times n$ matrix, $n\geq 2$.

*Every skew-symmetric non-zero real $5 \times 5$ matrix.

*Every skew-symmetric non-zero real $2 \times 2$ matrix.

 A: We will use the properties $\det A=\det(A^t)$ and $\det(-A)=(-1)^d\det A$, where $d$ is the dimension of the matrix. 


*

*We have $\det(A+I)=\det(A^t+I)=\det(I-A)$. Let $x$ be such that $(A-I)x=0$. Then 
$$\langle x,x\rangle=\langle x,Ax\rangle=\langle A^tx,x\rangle=-\langle Ax,x\rangle=-\langle x,x\rangle,$$
which proves that $x=0$. We have seen that if $(A-I)x=0$ then $x=0$, 
hence $A-I$ is invertible. This implies that $\det(I-A)\neq 0 $. Since 
$\det(I-A)=\det(A+I)$ we get that $\det(A+I)\neq 0$, which proves that $A+I$ is invertible.  

*Using the mentioned properties, for a skew-symmetrix matrix of odd dimension the determinant is $0$. 

*In dimension 2, a non-zero skew-symmetric matrix is of the form $A:=\pmatrix{0&x\\-x&0}$ where $x\neq 0$. Its determinant is $x^2\neq 0$ hence $A$ is necessarily invertible.
For the other even dimensions (say $d=2N$), we can construct a non-zero skew-symmetric matrix which is not invertible. Indeed, consider the block matrix defined by 
$$
\pmatrix{A&0_{2,2N-2}\\ 0_{2N-2,2}& 0_{2N-2,2N-2}             }  
$$
where $0_{i,j}$ denoted the matrix with $i$ rows and $j$ columns and all its entries are zero as $A=   \pmatrix{0&1\\-1&0}$.
