GO Classes DPP | GATE CS | Linear Algebra | Skew-symmetric Matrix

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GO Classes DPP | GATE CS | Linear Algebra | Skew-symmetric Matrix

GO Classes asked Apr 4

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legend_of_cse

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Given: $A$ is a $2027 \times 2027$ real skew-symmetric matrix.

Property: $A^T = -A$.

To find the determinant :

$\det(A^T) = \det(A)$
$\det(kA) = k^n \det(A)$, where $n$ is the order of the matrix.

$$\det(A^T) = \det(-A)$$

$$\det(A) = (-1)^{2027} \det(A)$$

Since $2027$ is an odd number:

$$\det(A) = -1 \cdot \det(A)$$

$$2\det(A) = 0 \implies \det(A) = 0$$

Correct Answer: C. 0

Skew-Symmetric Properties

1. Determinant of Odd-Order Skew-Symmetric Matrices

Property: The determinant of any odd-order skew-symmetric matrix is always 0.
Reasoning: shown in the solution above, if $n$ is odd, then $\det(A) = (-1)^n \det(A)$ leads to $\det(A) = -\det(A)$.

2. Determinant of Even-Order Skew-Symmetric Matrices

Property: The determinant of an even-order skew-symmetric matrix is always a perfect square.
Reasoning: This is derived from the Pfaffian, where $\det(A) = [Pf(A)]^2$. For a $2 \times 2$ matrix $\begin{pmatrix} 0 & a \\ -a & 0 \end{pmatrix}$, the determinant is $a^2$.

3. Main Diagonal Elements

Property: All principal diagonal elements of a skew-symmetric matrix must be zero.
Proof: By definition $a_{ij} = -a_{ji}$. For diagonal elements, $i=j$, so $a_{ii} = -a_{ii} \implies 2a_{ii} = 0 \implies a_{ii} = 0$.

4. Eigenvalues

Property: The eigenvalues of a real skew-symmetric matrix are either zero or purely imaginary.
Reasoning: Since the matrix is related to rotation and "anti-symmetry," its characteristic equation involves terms that do not allow for non-zero real roots. They always come in conjugate pairs (e.g., $0, i\lambda, -i\lambda$).

5. Trace of the Matrix

Property: $Tr(A) = 0$.
Reasoning: Since all diagonal elements are zero (Property 3), their sum (the trace) must be zero.

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Best Possible Answer
Given: $A$ is a $2027 \times 2027$ real skew-symmetric matrix.
Property: $A^T = -A$.
To find the determinant :
$\det(A^T) = \det(A)$
$\det(kA) = k^n \det(A)$, where $n$ is the order of the matrix.
$$\det(A^T) = \det(-A)$$
$$\det(A) = (-1)^{2027} \det(A)$$
Since $2027$ is an odd number:
$$\det(A) = -1 \cdot \det(A)$$
$$2\det(A) = 0 \implies \det(A) = 0$$
Correct Answer: C. 0
Skew-Symmetric Properties
1. Determinant of Odd-Order Skew-Symmetric Matrices
Property: The determinant of any odd-order skew-symmetric matrix is always 0.
Reasoning: shown in the solution above, if $n$ is odd, then $\det(A) = (-1)^n \det(A)$ leads to $\det(A) = -\det(A)$.

2. Determinant of Even-Order Skew-Symmetric Matrices
Property: The determinant of an even-order skew-symmetric matrix is always a perfect square.
Reasoning: This is derived from the Pfaffian, where $\det(A) = [Pf(A)]^2$. For a $2 \times 2$ matrix $\begin{pmatrix} 0 & a \\ -a & 0 \end{pmatrix}$, the determinant is $a^2$.

3. Main Diagonal Elements
Property: All principal diagonal elements of a skew-symmetric matrix must be zero.
Proof: By definition $a_{ij} = -a_{ji}$. For diagonal elements, $i=j$, so $a_{ii} = -a_{ii} \implies 2a_{ii} = 0 \implies a_{ii} = 0$.

4. Eigenvalues
Property: The eigenvalues of a real skew-symmetric matrix are either zero or purely imaginary.
Reasoning: Since the matrix is related to rotation and "anti-symmetry," its characteristic equation involves terms that do not allow for non-zero real roots. They always come in conjugate pairs (e.g., $0, i\lambda, -i\lambda$).

5. Trace of the Matrix
Property: $Tr(A) = 0$.
Reasoning: Since all diagonal elements are zero (Property 3), their sum (the trace) must be zero. — legend_of_cse, Apr 4
Note : One more note Determinant of Even order Skew symmetric matrix is always a perfect square . which will use in eliminating option sometime. — akash_kumar 9, Apr 4
Damn, thanks for that. I was wondering there must be more interesting properties to skew-symmetric matrices.

Check this out: [Pfaffian] — Meticulous_March, Apr 4

GO Classes · Answer 1 · 2026-04-04T03:08:35+0000

If $A$ were a $2027 \times 2027$ real skew-symmetric matrix, then $A^T=-A$.

Taking determinants, we get

$\det(A^T)=\det(-A)$.

Since $\det(A^T)=\det(A)$, this gives

$\det(A)=(-1)^{2027}\det(A)$.

Because $2027$ is odd, $(-1)^{2027}=-1$, so

$\det(A)=-\det(A)$.

Therefore,

$2\det(A)=0$,

which implies

$\det(A)=0$.

So if the matrix were $2027 \times 2027$, the determinant would be $\boxed{0}$.

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GO Classes DPP | GATE CS | Linear Algebra | Skew-symmetric Matrix

Best Possible Answer

Skew-Symmetric Properties

1. Determinant of Odd-Order Skew-Symmetric Matrices

2. Determinant of Even-Order Skew-Symmetric Matrices

3. Main Diagonal Elements

4. Eigenvalues

5. Trace of the Matrix

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Skew-Symmetric Properties

1. Determinant of Odd-Order Skew-Symmetric Matrices

2. Determinant of Even-Order Skew-Symmetric Matrices

3. Main Diagonal Elements

4. Eigenvalues

5. Trace of the Matrix

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