Q.If $A' = \begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix}$, then find $(A + 2B)'$

Step 1 — The definition and the tool

$A$ is skew symmetric means

The key property of the transpose we need is the reversal law $(AB)^{T}=B^{T}A^{T}$, which for a product of a matrix with itself gives

(Applying the reversal law $n-1$ times to $A\cdot A\cdots A$ just reverses the order of identical factors, leaving $A^{T}$ multiplied $n$ times.)

Step 2 — Transpose $A^{2021}$

A scalar factor pulls out of a matrix power as $(cA)^{n}=c^{n}A^{n}$, so with $c=-1$:

because $2021$ is odd, so $(-1)^{2021}=-1$.

Step 3 — Read the conclusion

This is precisely the definition of a skew symmetric matrix applied to $A^{2021}$.

General rule worth remembering: if $A$ is skew symmetric, then

• $A^{n}$ is skew symmetric when $n$ is odd, …

Step 1 — Check (A): "A row matrix has only one row."

True by definition. A row matrix (row vector) has order $1\times n$, e.g. $\begin{bmatrix}2 & 5 & 7\end{bmatrix}$. ✓

Step 2 — Check (B): "A diagonal matrix has all diagonal elements equal to zero."

False. The definition is: a square matrix $A=[a_{ij}]$ is diagonal if

That is, everything off the main diagonal is zero; the diagonal entries $a_{ii}$ are unrestricted. Counter-example:

is diagonal, yet both diagonal elements are $1$, not $0$. (A matrix with every entry zero is the null matrix, a special case — not the definition.) ✗

Step 3 — Check (C): "A symmetric matrix $A$ is a square matrix satisfying $A'=A$."

True — that is precisely the definition: $a_{ij}=a_{ji}$ for all $i,j$. ✓

Step 4 — Check (D): "A skew-symmetric matrix has all diagonal elements equal to zero." …