Which of the following is not correct?

Question:

Which of the following is not correct?

(a) $|A|=\left|A^{T}\right|$, where $A=\left[a_{i j}\right]_{3 \times 3}$

(b) $|k A|=\left|k^{3}\right|$, where $A=\left[a_{i j}\right]_{3 \times 3}$

(c) If $A$ is a skew-symmetric matrix of odd order, then $|A|=0$

(d) $\left|\begin{array}{ll}a+b & c+d \\ e+f & g+h\end{array}\right|=\left|\begin{array}{ll}a & c \\ e & g\end{array}\right|+\left|\begin{array}{ll}b & d \\ f & h\end{array}\right|$

Solution:

(d) $\left|\begin{array}{ll}a+b & c+d \\ e+f & g+h\end{array}\right|=\left|\begin{array}{ll}a & c \\ e & g\end{array}\right|+\left|\begin{array}{ll}b & d \\ f & h\end{array}\right|$

$\mid a+b c+d$

$e+f g+h|=| a+b c$

$e+f g|+| a+b d$

$e+f h \mid$

$=\mid a c$

$e \quad g|+| b c$

$f \quad g|+| a b$

$e \quad h|+| b d$

$f \quad h \mid$

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