91Ó°ÊÓ

Diagonalizable Matrices and Eigenvalues In Exercises \(1-6,\) (a) verify that \(A\) is diagonalizable by finding \(P^{-1} A P,\) and \((b)\) use the result of part (a) and Theorem 7.4 to find the eigenvalues of \(A .\) $$ A=\left[\begin{array}{rrr} -1 & 1 & 0 \\ 0 & 3 & 0 \\ 4 & -2 & 5 \end{array}\right], P=\left[\begin{array}{rrr} 0 & 1 & -3 \\ 0 & 4 & 0 \\ 1 & 2 & 2 \end{array}\right] $$

A population has the characteristics below. (a) A total of \(75 \%\) of the population survives the first year. Of that \(75 \%, 25 \%\) survives the second year. The maximum life span is 3 years. (b) The average number of offspring for each member of the population is 2 the first year, 4 the second year, and 2 the third year. The population now consists of 160 members in each of the three age classes. How many members will there be in each age class in 1 year? in 2 years?

Showing That a Matrix Is Not Diagonalizable In Exercises \(15-22,\) show that the matrix is not diagonalizable. $$ \left[\begin{array}{ll} 0 & 0 \\ 5 & 0 \end{array}\right] $$

Showing That a Matrix Is Not Diagonalizable In Exercises \(15-22,\) show that the matrix is not diagonalizable. $$ \left[\begin{array}{rr} 1 & \frac{1}{2} \\ -2 & -1 \end{array}\right] $$

Showing That a Matrix Is Not Diagonalizable In Exercises \(15-22,\) show that the matrix is not diagonalizable. $$ \left[\begin{array}{rrr} 1 & -2 & 1 \\ 0 & 1 & 4 \\ 0 & 0 & 2 \end{array}\right] $$

Determining a Sufficient Condition for Diagonalization In Exercises \(23-26,\) find the eigenvalues of the matrix and determine whether there is a sufficient number of eigenvalues to guarantee that the matrix is diagonalizable by Theorem 7.6. $$ \left[\begin{array}{rrr} -3 & -2 & 3 \\ 3 & 4 & -9 \\ 1 & 2 & -5 \end{array}\right] $$

In Exercises \(19-32,\) determine whether the matrix is orthogonal. If the matrix is orthogonal, then show that the column vectors of the matrix form an orthonormal set. $$\left[\begin{array}{rrr} -4 & 0 & 3 \\ 0 & 1 & 0 \\ 3 & 0 & 4 \end{array}\right]$$

Finding a Basis In Exercises \(27-30,\) find a basis \(B\) for the domain of \(T\) such that the matrix for \(T\) relative to \(B\) is diagonal. $$ \begin{aligned} &T: P_{2} \rightarrow P_{2}\\\ &T\left(c+b x+a x^{2}\right)=(3 c+a)+(2 b+3 a) x+a x^{2}\end{aligned} $$

In Exercises \(33-38,\) show that any two eigenvectors of the symmetric matrix corresponding to distinct eigenvalues are orthogonal. $$\left[\begin{array}{rr} -1 & -2 \\ -2 & 2 \end{array}\right]$$

Writing Can a matrix be similar to two different diagonal matrices? Explain.