Miscellaneous · Q10 (MCQ): Both symmetric and skew symmetric

Basics you need

Solution

1. A must be square

   The book defines symmetric (Definition 4) and skew symmetric (Definition 5) only for square matrices. So A is $n\times n$⁠. (This also makes sense directly: if A were $m\times n$ with $m \ne n$⁠, then $A'$ would be $n\times m$⁠, a different order, so $A' = A$ could not even hold.)

   We settle this first because one of the options is "A is a square matrix", and because the final answer must say which zero matrix: a square one.
   Definitions 4 and 5, Section 3.6. Lesson: symmetric and skew symmetric matrices

2. Combine the two conditions

   Both conditions give a formula for $A'$⁠, so the right sides are equal:

   $$\begin{aligned} A &= A' = -A \\ A + A &= O \\ 2A &= O \\ A &= O. \end{aligned}$$

   The last step multiplies both sides by the number $\frac12$⁠.

   From $A = -A$⁠, adding A to both sides gives $2A = O$ (because $-A + A = O$⁠). Multiplying by the scalar $\frac12$ multiplies every entry by $\frac12$⁠, and half of 0 is 0, so $A = \frac12 O = O$⁠.
   Properties of matrix addition and scalar multiplication, Sections 3.4.3 and 3.4.4. Lesson: addition and scalar multiplication

3. The same thing, entry by entry

   For every $i, j$⁠, symmetry gives $a_{ji} = a_{ij}$ and skew symmetry gives $a_{ji} = -a_{ij}$⁠. So

   $$\begin{aligned} a_{ij} &= -a_{ij} \\ 2a_{ij} &= 0 \\ a_{ij} &= 0 \end{aligned}$$

   for every position, on and off the diagonal.

   This is the same argument as Step 2, written for a single entry, and it shows why nothing survives: the diagonal entries are already 0 for any skew symmetric matrix, and the off-diagonal ones are killed by having to be equal and opposite to their mirror images.
   Definition 5 and the remark after it (Section 3.6): the diagonal of a skew symmetric matrix is 0. Basics: mirror patterns

4. The square zero matrix really is both

   Conversely, for the $n\times n$ zero matrix $O$⁠: every entry is 0, so $O' = O$ (symmetric) and $-O = O$⁠, so $O' = -O$ (skew symmetric). Hence:

   A is both symmetric and skew symmetric if and only if A is a square zero matrix.

   Steps 2 and 3 show "both" forces $A = O$⁠. This step shows the converse, so the description "zero matrix" is exact: it includes every such A and nothing else.
   Section 3.3 (vii): the zero matrix O. Lesson: types of matrices

5. Examine every option

   Option	Is it true of A?	Verdict
   (A)	Yes, but weak. A square zero matrix has all off-diagonal entries 0, so it is (technically) diagonal. But "diagonal" allows non-zero diagonal entries: the diagonal matrix with diagonal $1, 2$ is symmetric but not skew symmetric.	Not the answer: it does not describe A.
   (B)	Yes, and it is exact (Steps 2 to 4).	Correct.
   (C)	Yes, but very weak. Every symmetric matrix is square (Step 1). Most square matrices, such as the one with rows $(1, 2)$ and $(3, 4)$⁠, are neither symmetric nor skew symmetric.	Not the answer: it does not describe A.
   (D)	No: option (B) is a correct description.	Wrong.

   Strictly speaking, (A), (B) and (C) are all true statements about A. The question asks what A is, and only (B) answers that completely: it determines A, and it implies both (A) and (C), whereas neither (A) nor (C) implies (B). In an MCQ where several options are consequences, the intended answer is the strongest one, the one that characterises the object. The answer key agrees: (B).
   Types (iii), (iv) and (vii) in Section 3.3. Lesson: types of matrices

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