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MathMedAdv 2015 • Paper 1

Symmetry of Matrices Formed by Skew Symmetric and Symmetric Matrices

Let XX and YY be two arbitrary, 3×33 \times 3, non-zero, skew-symmetric matrices and ZZ be an arbitrary 3×33 \times 3, non-zero, symmetric matrix. Then which of the following matrices is (are) skew-symmetric?

Options

A

Y3Z4−Z4Y3Y^3 Z^4 - Z^4 Y^3

B

X44+Y44X^{44} + Y^{44}

C

X4Z3−Z3X4X^4 Z^3 - Z^3 X^4

Correct
D

X23+Y23X^{23} + Y^{23}

Correct

Topics & Concepts

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Step-by-Step Solution

To determine which of the given matrices is (are) skew-symmetric, we recall the definitions and properties of symmetric and skew-symmetric matrices:

  1. A matrix MM is skew-symmetric if MT=−MM^T = -M.
  2. A matrix MM is symmetric if MT=MM^T = M.
  3. For any matrices AA and BB, (A+B)T=AT+BT(A + B)^T = A^T + B^T and (AB)T=BTAT(AB)^T = B^T A^T.
  4. For an integer n≥1n \ge 1, (An)T=(AT)n(A^n)^T = (A^T)^n.

We are given that XX and YY are skew-symmetric matrices, and ZZ is a symmetric matrix: XT=−X,YT=−Y,ZT=ZX^T = -X, \quad Y^T = -Y, \quad Z^T = Z

Now, let us analyze each option individually:

Option (A):

Let M=Y3Z4−Z4Y3M = Y^3 Z^4 - Z^4 Y^3. Taking the transpose of MM: MT=(Y3Z4−Z4Y3)T=(Y3Z4)T−(Z4Y3)TM^T = (Y^3 Z^4 - Z^4 Y^3)^T = (Y^3 Z^4)^T - (Z^4 Y^3)^T Using the reversal law of transposition: MT=(Z4)T(Y3)T−(Y3)T(Z4)TM^T = (Z^4)^T (Y^3)^T - (Y^3)^T (Z^4)^T Since (Z4)T=(ZT)4=Z4(Z^4)^T = (Z^T)^4 = Z^4 and (Y3)T=(YT)3=(−Y)3=−Y3(Y^3)^T = (Y^T)^3 = (-Y)^3 = -Y^3, we have: MT=Z4(−Y3)−(−Y3)Z4=−Z4Y3+Y3Z4=Y3Z4−Z4Y3=MM^T = Z^4(-Y^3) - (-Y^3)Z^4 = -Z^4 Y^3 + Y^3 Z^4 = Y^3 Z^4 - Z^4 Y^3 = M Since MT=MM^T = M, the matrix is symmetric. Thus, option (A) is incorrect.

Option (B):

Let M=X44+Y44M = X^{44} + Y^{44}. Taking the transpose of MM: MT=(X44+Y44)T=(X44)T+(Y44)T=(XT)44+(YT)44M^T = (X^{44} + Y^{44})^T = (X^{44})^T + (Y^{44})^T = (X^T)^{44} + (Y^T)^{44} Substituting XT=−XX^T = -X and YT=−YY^T = -Y: MT=(−X)44+(−Y)44=X44+Y44=MM^T = (-X)^{44} + (-Y)^{44} = X^{44} + Y^{44} = M Since MT=MM^T = M, the matrix is symmetric. Thus, option (B) is incorrect.

Option (C):

Let M=X4Z3−Z3X4M = X^4 Z^3 - Z^3 X^4. Taking the transpose of MM: MT=(X4Z3−Z3X4)T=(X4Z3)T−(Z3X4)TM^T = (X^4 Z^3 - Z^3 X^4)^T = (X^4 Z^3)^T - (Z^3 X^4)^T Using the reversal law: MT=(Z3)T(X4)T−(X4)T(Z3)TM^T = (Z^3)^T (X^4)^T - (X^4)^T (Z^3)^T Since (Z3)T=(ZT)3=Z3(Z^3)^T = (Z^T)^3 = Z^3 and (X4)T=(XT)4=(−X)4=X4(X^4)^T = (X^T)^4 = (-X)^4 = X^4, we get: MT=Z3X4−X4Z3=−(X4Z3−Z3X4)=−MM^T = Z^3 X^4 - X^4 Z^3 = -(X^4 Z^3 - Z^3 X^4) = -M Since MT=−MM^T = -M, the matrix is skew-symmetric. Thus, option (C) is correct.

Option (D):

Let M=X23+Y23M = X^{23} + Y^{23}. Taking the transpose of MM: MT=(X23+Y23)T=(X23)T+(Y23)T=(XT)23+(YT)23M^T = (X^{23} + Y^{23})^T = (X^{23})^T + (Y^{23})^T = (X^T)^{23} + (Y^T)^{23} Substituting XT=−XX^T = -X and YT=−YY^T = -Y: MT=(−X)23+(−Y)23=−X23−Y23=−(X23+Y23)=−MM^T = (-X)^{23} + (-Y)^{23} = -X^{23} - Y^{23} = -(X^{23} + Y^{23}) = -M Since MT=−MM^T = -M, the matrix is skew-symmetric. Thus, option (D) is correct.

Conclusion:

The matrices in options (C) and (D) are skew-symmetric.

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