Matrices — JEE Advanced · 2015 · 24 May 2015 · Paper 1 Question with Solution

Let

X

and

Y

be two arbitrary,

3 \times 3

, non-zero, skew-symmetric matrices and

Z

be an arbitrary

3 \times 3

, non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?

Answer: (C), (D)

Solution

For skew-symmetric matrices,

A^{T} = - A

. For symmetric matrices,

B^{T} = B

.

(A)

(Y^{3} Z^{4} - Z^{4} Y^{3})^{T} = (Z^{4})^{T} (Y^{3})^{T} - (Y^{3})^{T} (Z^{4})^{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})

. This is skew-symmetric only if the matrices commute, which is not generally true. Actually, this simplifies to

Z^{4} Y^{3} - Y^{3} Z^{4}

, which is the negative only if they commute. Not generally skew-symmetric.

(B)

(X^{44} + Y^{44})^{T} = (X^{T})^{44} + (Y^{T})^{44} = (- X)^{44} + (- Y)^{44} = X^{44} + Y^{44}

. This is symmetric, not skew-symmetric.

(C)

(X^{4} Z^{3} - Z^{3} X^{4})^{T} = (Z^{3})^{T} (X^{4})^{T} - (X^{4})^{T} (Z^{3})^{T} = Z^{3} X^{4} - X^{4} Z^{3} = - (X^{4} Z^{3} - Z^{3} X^{4})

. This is skew-symmetric. ✓

(D)

(X^{23} + Y^{23})^{T} = (- X)^{23} + (- Y)^{23} = - X^{23} - Y^{23} = - (X^{23} + Y^{23})

. This is skew-symmetric. ✓

JEE Advanced · 2015 · 24 May 2015 · Paper 1 · One or more correct options