91Ó°ÊÓ

Calculate the square of the matrix by multiplying A by itself: \(A^2 = A \cdot A\).

Inspect the entries of \(A^2\). If all entries are positive, then matrix A is regular. If not, continue testing higher powers of the matrix until either a power with all positive entries is found or it is determined that no such power exists. We have checked higher powers for matrix (a) and found that no power has all positive entries. Therefore, matrix (a) is NOT regular. Follow the same process for the remaining matrices: (b) Matrix \(B = \left(\begin{array}{rrr}0.5 & 0 & 1 \\\ 0.5 & 0 & 0 \\\ 0 & 1 & 0\end{array}\right)\) Checking its powers, we find that matrix B is NOT regular. (c) Matrix \(C = \left(\begin{array}{rrr}0.5 & 0 & 0 \\\ 0.5 & 0 & 1 \\\ 0 & 1 & 0\end{array}\right)\) Checking its powers, we find that matrix C is NOT regular. (d) Matrix \(D = \left(\begin{array}{rrr}0.5 & 0 & 1 \\\ 0.5 & 1 & 0 \\\ 0 & 0 & 0\end{array}\right)\) Checking its powers, we find that matrix D is NOT regular. (e) Matrix \(E = \left(\begin{array}{ccc}\frac{1}{3} & 0 & 0 \\\ \frac{1}{3} & 1 & 0 \\\ \frac{1}{3} & 0 & 1\end{array}\right)\) Checking its powers, we find that matrix E IS regular. (f) Matrix \(F = \left(\begin{array}{rrr}1 & 0 & 0 \\\ 0 & 0.7 & 0.2 \\\ 0 & 0.3 & 0.8\end{array}\right)\) Checking its powers, we find that matrix F IS regular. (g) Matrix \(G = \left(\begin{array}{cccc}0 & \frac{1}{2} & 0 & 0 \\\ \frac{1}{2} & 0 & 0 & 0 \\\ \frac{1}{4} & \frac{1}{4} & 1 & 0 \\\ \frac{1}{4} & \frac{1}{4} & 0 & 1\end{array}\right)\) Checking its powers, we find that matrix G IS regular. (h) Matrix \(H = \left(\begin{array}{cccc}\frac{1}{4} & \frac{1}{4} & 0 & 0 \\\ \frac{1}{4} & \frac{1}{4} & 0 & 0 \\\ \frac{1}{4} & \frac{1}{4} & 1 & 0 \\\ \frac{1}{4} & \frac{1}{4} & 0 & 1\end{array}\right)\) Checking its powers, we find that matrix H is NOT regular. In summary, matrices (e), (f), and (g) are regular matrices, while the others are not.