Question: 18. Apply the result of Exercise 16 to find the determinants of the following matrices, and confirm your answers using a matrix program.

\(\left( {\begin{array}{*{20}{c}}{\bf{3}}&{\bf{8}}&{\bf{8}}&{\bf{8}}\\{\bf{8}}&{\bf{3}}&{\bf{8}}&{\bf{8}}\\{\bf{8}}&{\bf{8}}&{\bf{3}}&{\bf{8}}\\{\bf{8}}&{\bf{8}}&{\bf{8}}&{\bf{3}}\end{array}} \right)\) \(\left( {\begin{array}{*{20}{c}}{\bf{8}}&{\bf{3}}&{\bf{3}}&{\bf{3}}&{\bf{3}}\\{\bf{3}}&{\bf{8}}&{\bf{3}}&{\bf{3}}&{\bf{3}}\\{\bf{3}}&{\bf{3}}&{\bf{8}}&{\bf{3}}&{\bf{3}}\\{\bf{3}}&{\bf{3}}&{\bf{3}}&{\bf{8}}&{\bf{3}}\\{\bf{3}}&{\bf{3}}&{\bf{3}}&{\bf{3}}&{\bf{8}}\end{array}} \right)\)

From Exercise 16, you have \(\det A = {\left( {a - b} \right)^{n - 1}}\left( {a + \left( {n - 1} \right)b} \right)\) for \(A = \left( {\begin{array}{*{20}{c}}a&b&b& \cdots &b\\b&a&b& \cdots &b\\b&b&a& \cdots &b\\ \vdots & \vdots & \vdots & \ddots & \vdots \\b&b&b& \cdots &a\end{array}} \right)\) is size ofn.

Let \({A_1} = \left( {\begin{array}{*{20}{c}}3&8&8&8\\8&3&8&8\\8&8&3&8\\8&8&8&3\end{array}} \right)\). Here, \(a = 3,b = 8,\) and \(n = 4\). Using the formula in Exercise 16, you get:

\(\begin{array}{c}det{A_1} = {\left( {3 - 8} \right)^{4 - 1}}\left( {3 + \left( {4 - 1} \right)8} \right)\\ = {\left( { - 5} \right)^3}\left( {3 + 24} \right)\\ = - 125\left( {27} \right)\\\det {A_1} = - 3375\end{array}\)

Let \({A_2} = \left( {\begin{array}{*{20}{c}}8&3&3&3&3\\3&8&3&3&3\\3&3&8&3&3\\3&3&3&8&3\\3&3&3&3&8\end{array}} \right)\). Here, \(a = 8,b = 3,\) and \(n = 5\). Therefore, using the formula in Exercise 16, you get:

\(\begin{array}{c}det{A_2} = {\left( {8 - 3} \right)^{5 - 1}}\left( {8 + \left( {5 - 1} \right)3} \right)\\ = {\left( 5 \right)^4}\left( {8 + 12} \right)\\ = 625\left( {20} \right)\\\det {A_2} = 12500\end{array}\)