Engineering Mathematics III

Show that \(\begin{vmatrix}a+b+2c&a&b\\c&b+c+2a&b\\c&a&c+a+2b\end{vmatrix}=2(a+b+c)^3\).

Show by using the properties of determinants: \(\begin{vmatrix}1+a&1&1&1\\1&1+b&1&1\\1&1&1+c&1\\1&1&1&1+d\end{vmatrix}=abcd(1+1/a+1/b+1/c+1/d)\).

Show that \(\begin{vmatrix}(b+c)^2&a^2&a^2\\b^2&(c+a)^2&b^2\\c^2&c^2&(a+b)^2\end{vmatrix}=2abc(a+b+c)^3\).