If A and B are symmetric matrices of order 3 x 3 then the matrix 2AB - BA is:

Given: A and B are symmetric matrices of order 3 x 3.

For symmetric matrices: $A^T = A$AT=A and $B^T = B$BT=B

For any matrices: $(XY)^T = Y^T X^T$(XY)T=YTXT

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Let $M = 2AB - BA$M=2AB−BA

Taking transpose:

$$M^T = (2AB - BA)^T$$MT=(2AB−BA)T

$$M^T = (2AB)^T - (BA)^T$$MT=(2AB)T−(BA)T

$$M^T = 2(AB)^T - (BA)^T$$MT=2(AB)T−(BA)T

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Applying the transpose property:

$$M^T = 2B^T A^T - A^T B^T$$MT=2BTAT−ATBT

Since $A^T = A$AT=A and $B^T = B$BT=B:

$$M^T = 2BA - AB$$MT=2BA−AB

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For M to be symmetric, we need $M^T = M$MT=M:

$$2BA - AB = 2AB - BA$$2BA−AB=2AB−BA

$$3BA = 3AB$$3BA=3AB

$$BA = AB$$BA=AB

This is not true in general since matrix multiplication is not commutative.

Therefore, M is not symmetric.

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For M to be skew-symmetric, we need $M^T = -M$MT=−M:

$$2BA - AB = -(2AB - BA)$$2BA−AB=−(2AB−BA)

$$2BA - AB = -2AB + BA$$2BA−AB=−2AB+BA

$$2BA - AB = BA - 2AB$$2BA−AB=BA−2AB

$$BA = -AB$$BA=−AB

This is also not true in general.

Therefore, M is not skew-symmetric.

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The matrix $2AB - BA$2AB−BA is neither symmetric nor skew-symmetric.