Question

Which of the following is not correct?
(a) $\left|A\right|=\left|A^T\right|,\mathrm{where}A={\left[a_{ij}\right]}_{3\times 3}$

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

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

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

Answer 1

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

$$\begin{gathered} \left|a+bc+d \\ e+fg+h\right|=\left|a+bc \\ e+fg\right|+\left|a+bd \\ e+fh\right| \\ =\left|ac \\ eg\right|+\left|bc \\ fg\right|+\left|ad \\ eh\right|+\left|bd \\ fh\right| \end{gathered}$$