Determining a Sufficient Condition for Diagonalization In Exercises 2326, find the

Diagonalizable Matrices and Eigenvalues In Exercises 16, (a) verify that A is diagonalizable by finding P1AP, and (b) use the result of part (a) and Theorem 7.4 to find the eigenvalues of A. A = [ 11 3 36 10], P = [ 3 1 4 1]

Diagonalizable Matrices and Eigenvalues In Exercises 16, (a) verify that A is diagonalizable by finding P1AP, and (b) use the result of part (a) and Theorem 7.4 to find the eigenvalues of A. A = [ 1 1 3 5], P = [ 3 1 1 1]

Diagonalizable Matrices and Eigenvalues In Exercises 16, (a) verify that A is diagonalizable by finding P1AP, and (b) use the result of part (a) and Theorem 7.4 to find the eigenvalues of A. A = [ 3 2 2 2], P = [ 1 2 2 1]

Diagonalizable Matrices and Eigenvalues In Exercises 16, (a) verify that A is diagonalizable by finding P1AP, and (b) use the result of part (a) and Theorem 7.4 to find the eigenvalues of A. A = [ 4 2 5 3], P = [ 1 1 5 2]

Diagonalizable Matrices and Eigenvalues In Exercises 16, (a) verify that A is diagonalizable by finding P1AP, and (b) use the result of part (a) and Theorem 7.4 to find the eigenvalues of A. A = [ 1 0 4 1 3 2 0 0 5 ] , P = [ 0 0 1 1 4 2 3 0 2 ]

Diagonalizable Matrices and Eigenvalues In Exercises 16, (a) verify that A is diagonalizable by finding P1AP, and (b) use the result of part (a) and Theorem 7.4 to find the eigenvalues of A. A = [ 0.80 0.10 0.05 0.05 0.10 0.80 0.05 0.05 0.05 0.05 0.80 0.10 0.05 0.05 0.10 0.80] , P = [ 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 ]

Diagonalizing a Matrix In Exercises 714, find (if possible) a nonsingular matrix P such that P1AP is diagonal. Verify that P1AP is a diagonal matrix with the eigenvalues on the main diagonal. A = [ 6 2 3 1] (See Exercise 15, Section 7.1.)

Diagonalizing a Matrix In Exercises 714, find (if possible) a nonsingular matrix P such that P1AP is diagonal. Verify that P1AP is a diagonal matrix with the eigenvalues on the main diagonal. A = [ 1 4 1 2 1 4 0] (See Exercise 20 Section 7.1.)

Diagonalizing a Matrix In Exercises 714, find (if possible) a nonsingular matrix P such that P1AP is diagonal. Verify that P1AP is a diagonal matrix with the eigenvalues on the main diagonal. A = [ 2 0 0 2 3 1 3 2 2 ] (See Exercise 21, Section 7.1.)

Diagonalizing a Matrix In Exercises 714, find (if possible) a nonsingular matrix P such that P1AP is diagonal. Verify that P1AP is a diagonal matrix with the eigenvalues on the main diagonal. A = [ 3 0 0 2 0 2 1 2 0 ] (See Exercise 22, Section 7.1.)

Diagonalizing a Matrix In Exercises 714, find (if possible) a nonsingular matrix P such that P1AP is diagonal. Verify that P1AP is a diagonal matrix with the eigenvalues on the main diagonal. A = [ 1 2 6 2 5 6 2 2 3 ] (See Exercise 23, Section 7.1.)

Diagonalizing a Matrix In Exercises 714, find (if possible) a nonsingular matrix P such that P1AP is diagonal. Verify that P1AP is a diagonal matrix with the eigenvalues on the main diagonal. A = [ 3 3 1 2 4 2 3 9 5 ] (See Exercise 24, Section 7.1.)

Diagonalizing a Matrix In Exercises 714, find (if possible) a nonsingular matrix P such that P1AP is diagonal. Verify that P1AP is a diagonal matrix with the eigenvalues on the main diagonal. A = [ 1 1 1 0 2 0 0 1 2 ]

Diagonalizing a Matrix In Exercises 714, find (if possible) a nonsingular matrix P such that P1AP is diagonal. Verify that P1AP is a diagonal matrix with the eigenvalues on the main diagonal. . A = [ 2 4 0 0 4 4 0 0 4 ]

Showing That a Matrix Is Not Diagonalizable In Exercises 1522, show that the matrix is not diagonalizable. [ 0 5 0 0]

Showing That a Matrix Is Not Diagonalizable In Exercises 1522, show that the matrix is not diagonalizable. [ 1 2 1 2 1]

Showing That a Matrix Is Not Diagonalizable In Exercises 1522, show that the matrix is not diagonalizable. [ 7 0 7 7]

Showing That a Matrix Is Not Diagonalizable In Exercises 1522, show that the matrix is not diagonalizable. [ 1 2 0 1]

Showing That a Matrix Is Not Diagonalizable In Exercises 1522, show that the matrix is not diagonalizable. [ 1 0 0 2 1 0 1 4 2 ]

Showing That a Matrix Is Not Diagonalizable In Exercises 1522, show that the matrix is not diagonalizable. [ 3 0 0 2 2 0 2 3 2 ]

Showing That a Matrix Is Not Diagonalizable In Exercises 1522, show that the matrix is not diagonalizable. [ 1 0 2 0 0 1 0 2 1 0 2 0 1 1 2 2 ] (See Exercise 39, Section 7.1.)

Showing That a Matrix Is Not Diagonalizable In Exercises 1522, show that the matrix is not diagonalizable. [ 1 1 2 1 3 4 0 0 3 3 1 0 3 3 1 0 ] (See Exercise 40, Section 7.1.)

Determining a Sufficient Condition for Diagonalization In Exercises 2326, 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. [ 1 1 1 1]

Determining a Sufficient Condition for Diagonalization In Exercises 2326, 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. [ 2 5 0 2]

Determining a Sufficient Condition for Diagonalization In Exercises 2326, 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. [ 3 3 1 2 4 2 3 9 5 ]

Determining a Sufficient Condition for Diagonalization In Exercises 2326, 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. [ 4 0 0 3 1 0 2 1 2 ]

Finding a Basis In Exercises 2730, find a basis B for the domain of T such that the matrix for T relative to B is diagonal. T: R2R2: T(x, y) = (x + y, x + y)

Finding a Basis In Exercises 2730, find a basis B for the domain of T such that the matrix for T relative to B is diagonal. T: R3R3: T(x, y, z) = (2x + 2y 3z, 2x + y 6z, x 2y)

Finding a Basis In Exercises 2730, find a basis B for the domain of T such that the matrix for T relative to B is diagonal. T: P1P1: T(a + bx) = a + (a + 2b)x

Finding a Basis In Exercises 2730, find a basis B for the domain of T such that the matrix for T relative to B is diagonal. T: P2P2: T(c + bx + ax2) = (3c + a) + (2b + 3a)x + ax2

Proof Let A be a diagonalizable n n matrix and let P be an invertible n n matrix such that B = P1AP is the diagonal form of A. Prove that Ak = PBk P1, where k is a positive integer.

Let 1, 2, . . . , n be n distinct eigenvalues of an n n matrix A. Use the result of Exercise 31 to find the eigenvalues of Ak.

Finding a Power of a Matrix In Exercises 3336, use the result of Exercise 31 to find the power of A shown. A = [ 10 6 18 11], A6

Finding a Power of a Matrix In Exercises 3336, use the result of Exercise 31 to find the power of A shown. A = [ 1 2 3 0], A7

Finding a Power of a Matrix In Exercises 3336, use the result of Exercise 31 to find the power of A shown. A = [ 2 0 3 0 2 0 2 2 3 ] , A5

Finding a Power of a Matrix In Exercises 3336, use the result of Exercise 31 to find the power of A shown. A = [ 2 2 2 3 5 1 2 0 4 ] , A8

True or False? In Exercises 37 and 38, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) If A and B are similar n n matrices, then they always have the same characteristic polynomial equation. (b) The fact that an n n matrix A has n distinct eigenvalues does not guarantee that A is diagonalizable

True or False? In Exercises 37 and 38, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) If A is a diagonalizable matrix, then it has n linearly independent eigenvectors. (b) If an n n matrix A is diagonalizable, then it must have n distinct eigenvalues.

Are the two matrices similar? If so, find a matrix P such that B = P1AP. A = [ 1 0 0 0 2 0 0 0 3 ] B = [ 3 0 0 0 2 0 0 0 1 ]

Calculus For a real number x, you can define ex by the series ex = 1 + x + x 2 2! + x 3 3! + x 4 4! + . . . . In a similar way, for a square matrix X, you can define eX by the series eX = I + X + 1 2!X2 + 1 3!X3 + 1 4!X4 + . . . . Evaluate eX, where X is the square matrix shown. (a) X = [ 1 0 0 1] (b) X = [ 1 1 0 0] (c) X = [ 0 1 1 0] (d) X = [ 2 0 0 2]

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

Proof Prove that if matrix A is diagonalizable, then AT is diagonalizable.

Proof Prove that if matrix A is diagonalizable with n real eigenvalues 1, 2, . . . , n, then A = 12 . . . n.

Proof Prove that the matrix A = [ a c b d] is diagonalizable when 4bc < (a d)2 and is not diagonalizable when 4bc > (a d)2.

Guided Proof Prove that if the eigenvalues of a diagonalizable matrix A are all 1, then the matrix is equal to its inverse. Getting Started: To show that the matrix is equal to its inverse, use the fact that there exists an invertible matrix P such that D = P1AP, where D is a diagonal matrix with 1 along its main diagonal. (i) Let D = P1AP, where D is a diagonal matrix with 1 along its main diagonal. (ii) Find A in terms of P, P1, and D. (iii) Use the properties of the inverse of a product of matrices and the fact that D is diagonal to expand to find A1. (iv) Conclude that A1 = A.

Guided Proof Prove that nonzero nilpotent matrices are not diagonalizable. Getting Started: From Exercise 80 in Section 7.1, you know that 0 is the only eigenvalue of the nilpotent matrix A. Show that it is impossible for A to be diagonalizable. (i) Assume A is diagonalizable, so there exists an invertible matrix P such that P1AP = D, where D is the zero matrix. (ii) Find A in terms of P, P1, and D. (iii) Find a contradiction and conclude that nonzero nilpotent matrices are not diagonalizable.

Proof Prove that if A is a nonsingular diagonalizable matrix, then A1 is also diagonalizable.

CAPSTONE Explain how to determine whether an n n matrix A is diagonalizable using (a) similar matrices, (b) eigenvectors, and (c) distinct eigenvalues.

Showing That a Matrix Is Not Diagonalizable In Exercises 49 and 50, show that the matrix is not diagonalizable. [ 4 0 k 4], k 0

Showing That a Matrix Is Not Diagonalizable In Exercises 49 and 50, show that the matrix is not diagonalizable. [ 0 k 0 0], k 0

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. \(A=\left[\begin{array}{rr} -11 & 36 \\ -3 & 10 \end{array}\right], P=\left[\begin{array}{cc} -3 & -4 \\ -1 & -1 \end{array}\right]\) Text Transcription: P^-1 AP A=[-11 36 -3 10], P=[-3 -4 -1 -1]

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. \(A=\left[\begin{array}{rr} 1 & 3 \\ -1 & 5 \end{array}\right], P=\left[\begin{array}{ll} 3 & 1 \\ 1 & 1 \end{array}\right]\) Text Transcription: P^-1 AP A=[1 3 -1 5], P=[3 1 1 1]

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. \(A=\left[\begin{array}{ll} 4 & -5 \\ 2 & -3 \end{array}\right], P=\left[\begin{array}{ll} 1 & 5 \\ 1 & 2 \end{array}\right]\) Text Transcription: P^-1 AP A=[4 -5 2 -3], P=[1 5 1 2]

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. \(A=\left[\begin{array}{ll} 4 & -5 \\ 2 & -3 \end{array}\right], P=\left[\begin{array}{ll} 1 & 5 \\ 1 & 2 \end{array}\right]\) Text Transcription: P^-1 AP A=[4 -5 2 -3], P=[1 5 1 2]

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. \(A=\left[\begin{array}{rrr} -1 & 1 & 0 \\ 0 & 3 & 0 \\ 4 & -2 & 5 \end{array}\right], P=\left[\begin{array}{rrr} 0 & 1 & -3 \\ 0 & 4 & 0 \\ 1 & 2 & 2 \end{array}\right]\) Text Transcription: P^-1 AP A=[-1 1 0 0 3 0 4 -2 5], P=[0 1 -3 0 4 0 1 2 2]

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. \(A=\left[\begin{array}{llll} 0.80 & 0.10 & 0.05 & 0.05 \\ 0.10 & 0.80 & 0.05 & 0.05 \\ 0.05 & 0.05 & 0.80 & 0.10 \\ 0.05 & 0.05 & 0.10 & 0.80 \end{array}\right]\) Text Transcription: P^-1 AP A=[0.80 0.10 0.05 0.05 0.10 0.80 0.05 0.05 0.05 0.05 0.80 0.10 0.05 0.05 0.10 0.80]

In Exercises 7–14, find (if possible) a nonsingular matrix P such that \(P^{-1} A P\) is diagonal. Verify that \(P^{-1} A P\) is a diagonal matrix with the eigenvalues on the main diagonal. \(A=\left[\begin{array}{rr} 6 & -3 \\ -2 & 1 \end{array}\right]\) (See Exercise 15, Section 7.1.) Text Transcription: P^-1 AP A=[6 -3 -2 1]

In Exercises 7–14, find (if possible) a nonsingular matrix P such that \(P^{-1} A P\) is diagonal. Verify that \(P^{-1} A P\) is a diagonal matrix with the eigenvalues on the main diagonal. \(A=\left[\begin{array}{ll} \frac{1}{4} & \frac{1}{4} \\ \frac{1}{2} & 0 \end{array}\right]\) (See Exercise 20, Section 7.1.) Text Transcription: P^-1 AP A=[1/4 1/4 1/2 0]

In Exercises 7–14, find (if possible) a nonsingular matrix P such that \(P^{-1} A P\) is diagonal. Verify that \(P^{-1} A P\) is a diagonal matrix with the eigenvalues on the main diagonal. \(A=\left[\begin{array}{rrr} 2 & -2 & 3 \\ 0 & 3 & -2 \\ 0 & -1 & 2 \end{array}\right]\) (See Exercise 21, Section 7.1.) Text Transcription: P^-1 AP A=[2 -2 3 0 3 -2 0 -1 2]

In Exercises 7–14, find (if possible) a nonsingular matrix P such that \(P^{-1} A P\) is diagonal. Verify that \(P^{-1} A P\) is a diagonal matrix with the eigenvalues on the main diagonal. \(A=\left[\begin{array}{lll} 3 & 2 & 1 \\ 0 & 0 & 2 \\ 0 & 2 & 0 \end{array}\right]\) (See Exercise 22, Section 7.1.) Text Transcription: P^-1 AP A=[3 2 1 0 0 2 0 2 0]

In Exercises 7–14, find (if possible) a nonsingular matrix P such that \(P^{-1} A P\) is diagonal. Verify that \(P^{-1} A P\) is a diagonal matrix with the eigenvalues on the main diagonal. \(A=\left[\begin{array}{rrr} 1 & 2 & -2 \\ -2 & 5 & -2 \\ -6 & 6 & -3 \end{array}\right]\) (See Exercise 23, Section 7.1.) Text Transcription: P^-1 AP A=[1 2 -2 -2 5 -2 -6 6 -3]

In Exercises 7–14, find (if possible) a nonsingular matrix P such that \(P^{-1} A P\) is diagonal. Verify that \(P^{-1} A P\) is a diagonal matrix with the eigenvalues on the main diagonal. \(A=\left[\begin{array}{rrr} 3 & 2 & -3 \\ -3 & -4 & 9 \\ -1 & -2 & 5 \end{array}\right]\) (See Exercise 24, Section 7.1.) Text Transcription: P^-1 AP A=[3 2 -3 -3 -4 9 -1 -2 5]

In Exercises 7–14, find (if possible) a nonsingular matrix P such that \(P^{-1} A P\) is diagonal. Verify that \(P^{-1} A P\) is a diagonal matrix with the eigenvalues on the main diagonal. \(A=\left[\begin{array}{lll} 1 & 0 & 0 \\ 1 & 2 & 1 \\ 1 & 0 & 2 \end{array}\right]\) Text Transcription: P^-1 AP A=[1 0 0 1 2 1 1 0 2]

In Exercises 7–14, find (if possible) a nonsingular matrix P such that \(P^{-1} A P\) is diagonal. Verify that \(P^{-1} A P\) is a diagonal matrix with the eigenvalues on the main diagonal. \(A=\left[\begin{array}{lll} 2 & 0 & 0 \\ 4 & 4 & 0 \\ 0 & 4 & 4 \end{array}\right]\) Text Transcription: P^-1 AP A=[2 0 0 4 4 0 0 4 4]

In Exercises 15–22, show that the matrix is not diagonalizable. \(\left[\begin{array}{ll} 0 & 0 \\ 5 & 0 \end{array}\right]\) Text Transcription: A=[2 0 0 4 4 0 0 4 4]

In Exercises 15–22, show that the matrix is not diagonalizable. \(\left[\begin{array}{rr} 1 & \frac{1}{2} \\ -2 & -1 \end{array}\right]\) Text Transcription: [1 1/2 -2 -1]

In Exercises 15–22, show that the matrix is not diagonalizable. \(\left[\begin{array}{ll} 7 & 7 \\ 0 & 7 \end{array}\right]\) Text Transcription: [7 7 0 7]

In Exercises 15–22, show that the matrix is not diagonalizable. \(\left[\begin{array}{rr} 1 & 0 \\ -2 & 1 \end{array}\right]\) Text Transcription: [1 0 -2 1]

In Exercises 15–22, show that the matrix is not diagonalizable. \(\left[\begin{array}{rrr} 1 & -2 & 1 \\ 0 & 1 & 4 \\ 0 & 0 & 2 \end{array}\right]\) Text Transcription: [1 -2 1 0 1 4 0 0 2]

In Exercises 15–22, show that the matrix is not diagonalizable. \(\left[\begin{array}{rrr} 3 & 2 & -2 \\ 0 & -2 & 3 \\ 0 & 0 & -2 \end{array}\right]\) Text Transcription: [3 2 -2 0 -2 3 0 0 -2]

In Exercises 15–22, show that the matrix is not diagonalizable. \(\left[\begin{array}{rrrr} 1 & 0 & -1 & 1 \\ 0 & 1 & 0 & 1 \\ -2 & 0 & 2 & -2 \\ 0 & 2 & 0 & 2 \end{array}\right]\) (See Exercise 39, Section 7.1.) Text Transcription: [1 0 -1 1 0 1 0 1 -2 0 2 -2 0 2 0 2]

In Exercises 15–22, show that the matrix is not diagonalizable. \(\left[\begin{array}{rrrr} 1 & -3 & 3 & 3 \\ -1 & 4 & -3 & -3 \\ -2 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 \end{array}\right]\) (See Exercise 40, Section 7.1.) Text Transcription: [1 -3 3 3 -1 4 -3 -3 -2 0 1 1 1 0 0 0]

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. \(\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right]\) Text Transcription: [1 1 1 1]

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. \(\left[\begin{array}{ll} 2 & 0 \\ 5 & 2 \end{array}\right]\) Text Transcription: [2 0 5 2]

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. \(\left[\begin{array}{rrr} -3 & -2 & 3 \\ 3 & 4 & -9 \\ 1 & 2 & -5 \end{array}\right]\) Text Transcription: [-3 -2 3 3 4 -9 1 2 -5]

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. \(\left[\begin{array}{rrr} 4 & 3 & -2 \\ 0 & 1 & 1 \\ 0 & 0 & -2 \end{array}\right]\) Text Transcription: [4 3 -2 0 1 1 0 0 -2]

In Exercises 27–30, find a basis B for the domain of T such that the matrix for T relative to B is diagonal. \(T: R^{2} \rightarrow R^{2}: T(x, y)=(x+y, x+y)\) Text Transcription: T: R^2 rightarrow R^2: T(x, y)=(x+y, x+y)

In Exercises 27–30, find a basis B for the domain of T such that the matrix for T relative to B is diagonal. \(\begin{array}{l} T: R^{3} \rightarrow R^{3}: \\ T(x, y, z)=(-2 x+2 y-3 z, 2 x+y-6 z,-x-2 y) \end{array}\) Text Transcription: T: R^3 rightarrow R^3: T(x, y, z)=(-2 x+2 y-3 z, 2 x+y-6 z,-x-2 y)

In Exercises 27–30, find a basis B for the domain of T such that the matrix for T relative to B is diagonal. \(T: P_{1} \rightarrow P_{1}: T(a+b x)=a+(a+2 b) x\) Text Transcription: T: P_1 rightarrow P_1: T(a+bx)=a+(a+2b)x

In Exercises 27–30, find a basis B for the domain of T such that the matrix for T relative to B is diagonal. \(\begin{array}{l} T: P_{2} \rightarrow P_{2}: \\ T\left(c+b x+a x^{2}\right)=(3 c+a)+(2 b+3 a) x+a x^{2} \end{array}\) Text Transcription: T: P_2 rightarrow P_2: T(c+bx+ax^2)=(3c+a)+(2b+3a)x+ax^2

Let A be a diagonalizable \(n \times n\) matrix and let P be an invertible \(n \times n\) matrix such that \(B=P^{-1} A P\) is the diagonal form of A. Prove that \(A^{k}=P B^{k} P^{-1}\), where k is a positive integer. Text Transcription: n times n B=P^-1 AP A^k=P B^k P^-1

Let \(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\) be n distinct eigenvalues of an \(n \times n\) matrix A. Use the result of Exercise 31 to find the eigenvalues of \(A^{k}\). Text Transcription: lambda_1, lambda_2, ldots, lambda_n n times n A^k

In Exercises 33–36, use the result of Exercise 31 to find the power of A shown. \(A=\left[\begin{array}{rr} 10 & 18 \\ -6 & -11 \end{array}\right], A^{6}\) Text Transcription: A=[10 18 -6 -11], A^6

In Exercises 33–36, use the result of Exercise 31 to find the power of A shown. \(A=\left[\begin{array}{ll} 1 & 3 \\ 2 & 0 \end{array}\right], A^{7}\) Text Transcription: A=[1 3 2 0], A^{7}

In Exercises 33–36, use the result of Exercise 31 to find the power of A shown. \(A=\left[\begin{array}{lll} 2 & 0 & -2 \\ 0 & 2 & -2 \\ 3 & 0 & -3 \end{array}\right], A^{5}\) Text Transcription: A=[2 0 -2 0 2 -2 3 0 -3], A^5

In Exercises 33–36, use the result of Exercise 31 to find the power of A shown. \(A=\left[\begin{array}{rrr} 2 & 3 & -2 \\ -2 & -5 & 0 \\ -2 & -1 & 4 \end{array}\right], A^{8}\) Text Transcription: A=[2 3 -2 -2 -5 0 -2 -1 4], A^8

In Exercises 37 and 38, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) If A and B are similar \(n \times n\) matrices, then they always have the same characteristic polynomial equation. (b) The fact that an \(n \times n\) matrix A has n distinct eigenvalues does not guarantee that A is diagonalizable. Text Transcription: n times n

In Exercises 37 and 38, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) If A is a diagonalizable matrix, then it has n linearly independent eigenvectors. (b) If an \(n \times\) n matrix A is diagonalizable, then it must have n distinct eigenvalues. Text Transcription: n times n

Are the two matrices similar? If so, find a matrix P such that \(B=P^{-1} A P\). \(A=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{array}\right]\) \(B=\left[\begin{array}{lll} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{array}\right]\) Text Transcription: B=P^-1 AP A=[1 0 0 0 2 0 0 0 3] B=[3 0 0 0 2 0 0 0 1]

Calculus For a real number x, you can define ex by the series ex = 1 + x + x 2 2! + x 3 3! + x 4 4! + . . . . In a similar way, for a square matrix X, you can define eX by the series eX = I + X + 1 2!X2 + 1 3!X3 + 1 4!X4 + . . . . Evaluate eX, where X is the square matrix shown. (a) X = [ 1 0 0 1] (b) X = [ 1 1 0 0] (c) X = [ 0 1 1 0] (d) X = [ 2 0 0 2]

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

Prove that if matrix A is diagonalizable, then \(A^{T}\) is diagonalizable. Text Transcription: A^T

Prove that if matrix A is diagonalizable with n real eigenvalues \(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\), then \(|A|=\lambda_{1} \lambda_{2} \cdots \lambda_{n}\). Text Transcription: lambda_1, lambda_2, ldots, lambda_n |A|=lambda_1 lambda_2 cdots lambda_n

Prove that the matrix \(A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\) is diagonalizable when \(-4 b c<(a-d)^{2}\) and is not diagonalizable when \(-4 b c>(a-d)^{2}\). Text Transcription: A=[a b c d] -4 bc<(a-d)^2 -4 bc>(a-d)^2

Prove that if the eigenvalues of a diagonalizable matrix A are all \(\pm 1\), then the matrix is equal to its inverse. Getting Started: To show that the matrix is equal to its inverse, use the fact that there exists an invertible matrix P such that \(D=P^{-1} A P\), where D is a diagonal matrix with \(\pm 1\) along its main diagonal. (i) Let \(D=P^{-1} A P\), where D is a diagonal matrix with ±1 along its main diagonal. (ii) Find A in terms of P, \(P^{-1}\), and D. (iii) Use the properties of the inverse of a product of matrices and the fact that D is diagonal to expand to find \(A^{-1}\). (iv) Conclude that \(A^{-1}=A\). Text Transcription: pm 1 D=P^-1 AP P^-1 A^-1 A^-1=A

Prove that nonzero nilpotent matrices are not diagonalizable. Getting Started: From Exercise 80 in Section 7.1, you know that 0 is the only eigenvalue of the nilpotent matrix A. Show that it is impossible for A to be diagonalizable. (i) Assume A is diagonalizable, so there exists an invertible matrix P such that \(P^{-1} A P=D\), where D is the zero matrix. (ii) Find A in terms of P, \(P^{-1}\), and D. (iii) Find a contradiction and conclude that nonzero nilpotent matrices are not diagonalizable. Text Transcription: P^-1 AP=D P^-1

Prove that if A is a nonsingular diagonalizable matrix, then \(A^{-1}\) is also diagonalizable. Text Transcription: A^-1

Explain how to determine whether an \(n \times n\) matrix A is diagonalizable using (a) similar matrices, (b) eigenvectors, and (c) distinct eigenvalues. Text Transcription: n times n

In Exercises 49 and 50, show that the matrix is not diagonalizable. \(\left[\begin{array}{ll} 4 & k \\ 0 & 4 \end{array}\right], k \neq 0\) Text Transcription: [4 k 0 4], k neq 0

Showing That a Matrix Is Not Diagonalizable In Exercises 49 and 50, show that the matrix is not diagonalizable. [ 0 k 0 0], k 0