1. Algebra — Matrices and Determinants

Skew-symmetric matrices

A square matrix $A$ is termed skew-symmetric if its transpose equals its negative, i.e., $A^T = -A$, forcing all its principal diagonal entries to be strictly zero.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

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Definition and statement

Fundamental Decomposition Theorem for Square Matrices

Every square matrix $A$ can be uniquely expressed as the sum of a symmetric matrix $S$ and a skew-symmetric matrix $K$, given by $$A = \frac{1}{2}(A + A^T) + \frac{1}{2}(A - A^T)$$, where $S = \frac{1}{2}(A + A^T)$ is symmetric and $K = \frac{1}{2}(A - A^T)$ is skew-symmetric.

📐 Geometric Interpretation

In a linear transformation, a skew-symmetric matrix governs the rotational or infinitesimal orthogonal component (antisymmetric tensor field), leaving orthogonal invariants and generating rotations through matrix exponentials.

⚙️ Algebraic & Structural Meaning

The trace of any skew-symmetric matrix is zero, and its determinant for any odd dimension $n \times n$ is identically zero (i.e., $\det(A) = 0$ when $n$ is odd).

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Square Matrix Requirement	`Dimension $n \times n$`	Strictly Mandatory / Necessary	Transpose $A^T$ does not share the same domain and codomain dimensions for rectangular matrices, making $A = -A^T$ undefined.
Zero Principal Diagonal Elements	`Entries $a_{ii} = 0 \quad \forall i`	Strictly Mandatory	If any diagonal entry $a_{ii} \neq 0$, then $a_{ii} = -a_{ii} \implies 2a_{ii} = 0 \implies a_{ii} = 0$, leading to an immediate contradiction.
Off-diagonal Anti-symmetry	`Entries $a_{ij} = -a_{ji} \quad \forall i \neq j$`	Essential for definition	Failing this condition means the matrix is neither purely symmetric nor skew-symmetric, invalidating structural eigenvalue theorems.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Determinant of Odd-Order Skew-Symmetric Matrix

$$\det(A) = 0 \quad \text{if } n \text{ is odd}$$

Rule: If a skew-symmetric matrix has an odd order (e.g., $3 \times 3, 5 \times 5$), its determinant is instantly zero without computation.

Eigenvalues of Skew-Symmetric Matrices

$$\text{Eigenvalues of } A \text{ are either } 0 \text{ or purely imaginary } (\pm i\beta)$$

Rule: Real parts of all eigenvalues of a real skew-symmetric matrix are identically zero.

Trace Property

$$ \text{Tr}(A) = \sum_{i=1}^{n} a_{ii} = 0 $$

Rule: Since all diagonal elements are zero, the trace of any skew-symmetric matrix is always zero.

Application: Used for quick elimination in matrix property options.

Power Properties of Skew-Symmetric Matrices

$$A^{2k} \text{ is symmetric, } A^{2k+1} \text{ is skew-symmetric}$$

Rule: Even powers of a skew-symmetric matrix are symmetric, and odd powers are skew-symmetric.

Application: Asked in high-level UP PGT problems involving matrix polynomials.

Inverse of Skew-Symmetric Matrix (Even Order)

$$A^{-1} \text{ is skew-symmetric if } \det(A) \neq 0$$

Rule: If an even-order skew-symmetric matrix is invertible, its inverse is also skew-symmetric.

Application: Standard assertion-reason question pattern.

Rapid Elimination via Diagonal Check

$$a_{11} = a_{22} = \dots = a_{nn} = 0$$

Rule: Instantly reject any matrix option claiming to be skew-symmetric if any diagonal entry is non-zero.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Assuming Odd-Order Determinant is Non-Zero

Common Pitfall: Candidates spend valuable time calculating the determinant of a $3 \times 3$ skew-symmetric matrix.

Counterexample: For $A = \begin{pmatrix} 0 & 2 & -3 \\ -2 & 0 & 5 \\ 3 & -5 & 0 \end{pmatrix}$, $\det(A) = 0$

Why it fails: By skew-symmetry, $\det(A) = \det(A^T) = \det(-A) = (-1)^n \det(A)$. For odd $n$, $\det(A) = -\det(A) \implies 2\det(A) = 0 \implies \det(A) = 0$.

🛡️ Exam Defense: Never compute determinants of odd-order skew-symmetric matrices; mark 0 instantly.

⚠️ COMMON ERROR #02

Non-Zero Diagonal Entries in Options

Common Pitfall: Accepting matrices with non-zero entries on the main diagonal as skew-symmetric.

Counterexample: $A = \begin{pmatrix} 2 & 3 \\ -3 & 0 \end{pmatrix}$

Why it fails: $a_{11} = 2 \neq 0$, violating the fundamental necessity condition $A = -A^T$.

🛡️ Exam Defense: Scan the principal diagonal first before checking off-diagonal elements.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Symmetric Matrix	$A = A^T$	$a_{ij} = a_{ji}$	Reflection symmetry across main diagonal	All eigenvalues are real
Skew-Symmetric Matrix	$A = -A^T$	$a_{ii} = 0, a_{ij} = -a_{ji}$	Antisymmetric rotation / tensor	Diagonal is zero, odd order det is 0
Orthogonal Matrix	$A A^T = I$	$A^{-1} = A^T$	Rigid rotation / reflection (preserves length)	Determinant is $\pm 1$
Skew-Hermitian Matrix	$A^H = -A$	$a_{ii} = i \mathbb{R}, a_{ij} = -\bar{a}_{ji}$	Complex analogue of skew-symmetry	Purely imaginary diagonal entries

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

If $A$ is a skew-symmetric matrix of order $3$, what is the value of $\det(A)?$

(A) 1(B) -1(C) 0(D) \det(A)^3

📚 Method A: Formal Textbook Derivation

Let $A$ be of order $n$. Then $\det(A^T) = \det(-A) = (-1)^n \det(A)$. Since $A = A^T$, $\det(A) = (-1)^n \det(A)$. For $n = 3$ (odd), $\det(A) = -\det(A) \implies 2\det(A) = 0 \implies \det(A) = 0$.

⚡ Method B: 25-Second Exam Shortcut

Odd order skew-symmetric matrix determinant is always zero. Mark (C) in 2 seconds.

💡 Tip: Do not expand along rows; use the structural property directly.

CANONICAL PROBLEM 02 Dual-Method Breakdown

If $A = \begin{pmatrix} 0 & 2x & -2 \\ y & 0 & -4 \\ 2 & 4 & 0 \end{pmatrix}$ is a skew-symmetric matrix, find the values of $x$ and $y$.

(A) $x = -1/2, y = 4$(B) $x = -1, y = 2$(C) $x = 1/2, y = -4$(D) $x = 2, y = -2$

📚 Method A: Formal Textbook Derivation

For skew-symmetry, $a_{ij} = -a_{ji}$. Here $a_{12} = 2x$ and $a_{21} = y$, so $2x = -y$. Also $a_{13} = -2$ and $a_{31} = 2$ (satisfies $-2 = -2$). Check $a_{23} = -4$ and $a_{32} = 4$ (satisfies $-4 = -(-4)$). Thus $y = -2(2x) = -4x$. If $y = -4$, then $x = 1/2$.

⚡ Method B: 25-Second Exam Shortcut

Use $a_{ij} + a_{ji} = 0$. Hence $y + 2x = 0 \implies y = -2x$. Test options: only (C) gives $y = -4, x = 1/2$.

💡 Tip: Quickly verify cross-diagonal pairs sum to zero.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: If $A$ is a square matrix, then $A - A^T$ is always a:

A Symmetric matrix

B Skew-symmetric matrix

C Identity matrix

D Diagonal matrix

Derivation & Analysis:

Let $B = A - A^T$. Then $B^T = (A - A^T)^T = A^T - (A^T)^T = A^T - A = -(A - A^T) = -B$. Thus $B$ is skew-symmetric.

MCQ 2: The determinant of any skew-symmetric matrix of odd order is:

A Non-zero

B One

C Zero

D It depends on the order

Derivation & Analysis:

For an odd order $n$, $\det(A) = \det(A^T) = \det(-A) = (-1)^n \det(A) = -\det(A)$, which implies $\det(A) = 0$.

MCQ 3: If $A$ and $B$ are skew-symmetric matrices of the same order, then $AB$ is symmetric if:

A $A=-B$

B $A$ and $B$ are equal

C $AB=BA$

D $A$ or $B$ is the zero matrix

Derivation & Analysis:

$(AB)^T = B^T A^T = (-B)(-A) = BA$. For $AB$ to be symmetric, $(AB)^T = AB$, which requires $BA = AB$, meaning they commute under multiplication.

MCQ 4: All diagonal elements of a skew-symmetric matrix are:

A Zero

B One

C Arbitrary

D Equal to the corresponding off-diagonal entries

Derivation & Analysis:

For skew-symmetric $A$, $a_{ii} = -a_{ii}$, which implies $2a_{ii} = 0 \implies a_{ii} = 0$ for all $i$.

MCQ 5: If $A$ is a skew-symmetric matrix, then $A^{2n}$ (where $n \in \mathbb{N}$) is a:

A Skew-symmetric

B Symmetric

C Orthogonal

D Singular

Derivation & Analysis:

$(A^{2n})^T = (A^T)^{2n} = (-A)^{2n} = (-1)^{2n} A^{2n} = A^{2n}$, hence it is symmetric.

MCQ 6: If $A = \begin{pmatrix} 0 & 1 & -2 \\ -1 & 0 & 3 \\ 2 & -3 & 0 \end{pmatrix}$, then $A$ is:

A Symmetric

B Skew-symmetric

C Orthogonal

D Diagonal

Derivation & Analysis:

Principal diagonal elements are zero and $a_{ij} = -a_{ji}$ for all $i \neq j$, so $A^T = -A$, confirming it is skew-symmetric.

MCQ 7: The eigenvalues of a real skew-symmetric matrix are:

A All are positive real

B All are non-zero real

C Zero or purely imaginary

D All are equal

Derivation & Analysis:

Real skew-symmetric matrices have eigenvalues that are either zero or purely imaginary numbers.

MCQ 8: If $A$ is any square matrix, then $A + A^T$ is:

A Skew-symmetric

B Symmetric

C Zero

D Orthogonal

Derivation & Analysis:

$(A + A^T)^T = A^T + (A^T)^T = A^T + A = A + A^T$, which proves $A + A^T$ is symmetric.

MCQ 9: If $A$ is a skew-symmetric matrix of order $2$, then $\det(A)$ can be expressed as:

A $a_{12}^2$

B $-a_{12}^2$

C $0$

D $|a_{12}|$

Derivation & Analysis:

For order 2, $A = \begin{pmatrix} 0 & a \\ -a & 0 \end{pmatrix}$. Determinant is $0 - (-a^2) = a^2 = a_{12}^2$.

Q10

MCQ 10: Which of the following matrices cannot be skew-symmetric?

A $\begin{pmatrix}0&1\\-1&0\end{pmatrix}$

B $\begin{pmatrix}1&0\\0&1\end{pmatrix}$

C $\begin{pmatrix}0&-2\\2&0\end{pmatrix}$

D $\begin{pmatrix}0&3\\-3&0\end{pmatrix}$

Derivation & Analysis:

A skew-symmetric matrix satisfies $A^T=-A$, so every diagonal entry must be zero. Option (B) has non-zero diagonal entries and is not skew-symmetric.

Common questions

No. By definition, $A = -A^T$, which implies $a_{ii} = -a_{ii}$ for every diagonal entry. This forces $2a_{ii} = 0$, meaning every principal diagonal element must be strictly zero in the real field.

Students often waste time trying to calculate the determinant of odd-order skew-symmetric matrices or fail to recognize that any square matrix can be uniquely split into symmetric and skew-symmetric components.

Memorize core structural rules: odd-order skew-symmetric determinants are always zero, diagonal entries are always zero, and even powers of skew-symmetric matrices yield symmetric matrices.

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