Which of the following statements are correct?

(A) Inverse of a matrix, if it exists, is unique

(B) $(kA)' = -kA'$ (where k is any real number)

(C) For an invertible matrix $A$, $(A^{-1})^{-1} = A$

(D) For an invertible matrix $A$, $(A')^{-1} = (A^{-1})'$

Choose the correct answer from the options given below:

Statement (A): Inverse of a matrix, if it exists, is unique

A matrix can have only one inverse, not multiple inverses.

Assume matrix $A$ has two inverses $B$ and $C$.

$B = B \cdot I$

$B = B \cdot (A \cdot C)$

$B = (B \cdot A) \cdot C$

$B = I \cdot C$

$B = C$

Both inverses are the same. Hence, the inverse is unique.

Statement (A) is correct.

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Statement (B): $(kA)' = -kA'$ (where $k$ is any real number)

The correct formula is $(kA)' = kA'$ (positive $k$, not negative).

Consider $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $k = 5$

$kA = \begin{bmatrix} 5 & 10 \\ 15 & 20 \end{bmatrix}$

$(kA)' = \begin{bmatrix} 5 & 15 \\ 10 & 20 \end{bmatrix}$

$kA' = 5 \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$

$kA' = \begin{bmatrix} 5 & 15 \\ 10 & 20 \end{bmatrix}$

Therefore $(kA)' = kA'$, not $-kA'$.

Statement (B) is incorrect.

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Statement (C): For an invertible matrix $A$, $(A^{-1})^{-1} = A$

Let $B = A^{-1}$

By definition: $A \cdot B = I$ and $B \cdot A = I$

This means $A$ is the inverse of $B$.

$A = B^{-1}$

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

Statement (C) is correct.

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Statement (D): For an invertible matrix $A$, $(A')^{-1} = (A^{-1})'$

To show $(A^{-1})'$ is the inverse of $A'$:

$A' \cdot (A^{-1})' = (A^{-1} \cdot A)'$

$A' \cdot (A^{-1})' = I'$

$A' \cdot (A^{-1})' = I$

Since $A' \cdot (A^{-1})' = I$, the expression $(A^{-1})'$ is the inverse of $A'$.

Therefore $(A')^{-1} = (A^{-1})'$.

Statement (D) is correct.

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Statements (A), (C) and (D) are correct.