Q.(i) For what value of $x$ is the matrix $A=\begin{pmatrix} 0 & 1 & -2 \\ -1 & 0 & x^3 \\ 2 & -3 & 0\end{pmatrix}$ skew-symmetric?

Step 1. (i) Check the diagonal is already zero. In $A$, the diagonal entries are $0,0,0$, so that necessary condition is already met.

Step 2. (i) Check the $(1,2)$–$(2,1)$ pair. $a_{12}=1,\ a_{21}=-1$. Since $a_{21}=-a_{12}$ ($-1=-1$), this pair is already consistent — no condition on $x$ from here.

Step 3. (i) Check the $(1,3)$–$(3,1)$ pair. $a_{13}=-2,\ a_{31}=2$. Since $a_{31}=-a_{13}$ ($2=-(-2)$), this pair is also already consistent.

Step 4. (i) Use the $(2,3)$–$(3,2)$ pair to find $x$. $a_{23}=x^3,\ a_{32}=-3$. Skew-symmetry requires $a_{32}=-a_{23}$, i.e. $-3=-x^3 \Rightarrow x^3=3 \Rightarrow x=\sqrt[3]{3}$.

Step 5. (ii) Force the diagonal to zero. The $(2,2)$ entry is $q^2$; for skew-symmetry the diagonal must vanish, so $q^2=0 \Rightarrow q=0$. …

• Forgetting that a skew-symmetric matrix must have an all-zero diagonal (missing the $q^2=0$ condition) …