If A and B are invertible matrices of same order, then which one of the following is NOT true?

Two invertible matrices A and B of the same order are given. The task is to identify which property is NOT true.

"Invertible" means the matrix has an inverse.

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For $(AB)^{-1} = B^{-1}A^{-1}$:

This is the reverse order rule for matrix inverses.

When multiplying $(AB)(B^{-1}A^{-1})$, the B and B⁻¹ cancel, then A and A⁻¹ cancel, giving the identity matrix I.

This property is TRUE.

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For $adj(AB) = (adj A)(adj B)$:

The correct formula is:

$adj(AB) = (adj B)(adj A)$

The order is reversed, similar to matrix inverses.

The adjoint is related to inverse by: $A^{-1} = \frac{adj(A)}{|A|}$

Since inverses follow reverse order, adjoints follow the same pattern.

This property is NOT TRUE.

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For $adj(A^T) = (adj A)^T$:

The adjoint of transpose equals the transpose of adjoint.

This is a standard property where the operations are interchangeable.

This property is TRUE.

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For $|A^{-1}| = (|A|)^{-1}$:

The determinant of the inverse equals the reciprocal of the determinant.

From $A \cdot A^{-1} = I$:

$|A| \cdot |A^{-1}| = |I|$

$|A| \cdot |A^{-1}| = 1$

$|A^{-1}| = \frac{1}{|A|}$

$|A^{-1}| = (|A|)^{-1}$

This property is TRUE.

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The property that is NOT TRUE is $adj(AB) = (adj A)(adj B)$ because the adjoint of a product follows reverse order: $adj(AB) = (adj B)(adj A)$.