Let and be two arbitrary, , non-zero, skew-symmetric matrices and be an arbitrary , non-zero, symmetric matrix. Then which of the following matrices is (are) skew-symmetric? [2015]

Question

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? [2015]

Smart Achievers · Accepted Answer

(3, 4)

$X' = - X, Y' = - Y, Z' = Z$

$(Y^{3} Z^{4} - Z^{4} Y^{3})' = (Z^{4})' (Y^{3})' - (Y^{3})' (Z^{4})'$

$= {(Z')}^{4} {(Y')}^{3} - {(Y')}^{3} {(Z')}^{4}$

$= - Z^{4} Y^{3} + Y^{3} Z^{4} = Y^{3} Z^{4} - Z^{4} Y^{3}$

Therefore, $(Y^{3} Z^{4} - Z^{4} Y^{3})$ is a symmetric matrix.

Similarly, $X^{44} + Y^{44}$ is a symmetric matrix, and $X^{4} Z^{3} - Z^{3} X^{4}$ and $X^{23} + Y^{23}$ are skew-symmetric matrices.

Solution

Q.
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 [2015]

1 $Y^{3} Z^{4} - Z^{4} Y^{3}$

2 $X^{44} + Y^{44}$

3 $X^{4} Z^{3} - Z^{3} X^{4}$

4 $X^{23} + Y^{23}$

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Solution

Q. Let X and Y be two arbitrary, 3×3, non-zero, skew-symmetric matrices and Z be an arbitrary 3×3, non-zero, symmetric matrix. Then which of the following matrices is (are) skew-symmetric [2015]

1 Y3Z4-Z4Y3

2 X44+Y44

3 X4Z3-Z3X4

4 X23+Y23