For the linear programming problem(LPP), Maximize Z = 4x + y x + y ≤ 5 3x + y ≤ 9 x,y ≥ 0. Which of the following are NOT true? (A) The given LPP has…

Let $P = AB^T - BA^T$P=ABT−BAT

To determine the nature of this matrix, we find $P^T$PT and compare it with $P$P.

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$$P^T = (AB^T - BA^T)^T$$PT=(ABT−BAT)T

$$P^T = (AB^T)^T - (BA^T)^T$$PT=(ABT)T−(BAT)T

Using $(XY)^T = Y^TX^T$(XY)T=YTXT:

$$P^T = (B^T)^TA^T - (A^T)^TB^T$$PT=(BT)TAT−(AT)TBT

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Using $(X^T)^T = X$(XT)T=X:

$$P^T = BA^T - AB^T$$PT=BAT−ABT

$$P^T = -(AB^T - BA^T)$$PT=−(ABT−BAT)

$$P^T = -P$$PT=−P

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Since $P^T = -P$PT=−P, the matrix $(AB^T - BA^T)$(ABT−BAT) is always a skew-symmetric matrix.