Prove that if matrix A is diagonalizable with in real eigenvalues A, A2, There exists an...

Transcribed Image Text:

Prove that if matrix A is diagonalizable with in real eigenvalues A₁, A2, There exists an invertible matrix P and a diagonal matrix D, such that O Determinant of a Matrix Product Definition of the Inverse of a Matrix Properties of the Identity Matrix Determinant of a Triangular Matrix Determinant of an Inverse Matrix Definition of a Diagonalizable Matrix Eigenvalues of Triangular Matrices Associative Property of Matrix Multiplication Similar Matrices Have the Same Eigenvalues The eigenvalues of D are A₁, A₂, Determinant of a Matrix Product Definition of the Inverse of a Matrix Properties of the Identity Matrix Determinant of a Triangular Matrix Determinant of an Inverse Matrix Definition of a Diagonalizable Matrix Eigenvalues of Triangular Matrices Associative Property of Matrix Multiplication Similar Matrices Have the Same Eigenvalues The diagonal elements of D are A₁, A₂, A Determinant of a Matrix Product Definition of the Inverse of a Matrix Ⓒ Properties of the Identity Matrix Determinant of a Triangular Matrix O Determinant of an Inverse Matrix Definition of a Diagonalizable Matrix Eigenvalues of Triangular Matrices Athen |A|A₁A₂A. Complete the proof by justifying each step. P-1AP= D. |A| = |IAI Determinant of a Matrix Product Definition of the Inverse of a Matrix Properties of the Identity Matrix Determinant of a Triangular Matrix Determinant of an Inverse Matrix Definition of a Diagonalizable Matrix Eigenvalues of Triangular Matrices Associative Property of Matrix Multiplication Similar Matrices Have the Same Eigenvalues |A| = |(PP-1)A(PP-1)| Determinant of a Matrix Product Definition of the Inverse of a Matrix Properties of the Identity Matrix O Determinant of a Triangular Matrix Determinant of an Inverse Matrix Ⓒ Definition of a Diagonalizable Matrix Ⓒ Eigenvalues of Triangular Matrices Associative Property of Matrix Multiplication Similar Matrices Have the Same Eigenvalues |A| = |P(P-¹AP) P-1| = |PDP-11 Determinant of a Matrix Product Definition of the Inverse of a Matrix Properties of the Identity Matrix Determinant of a Triangular Matrix O Determinant of an Inverse Matrix Definition of a Diagonalizable Matrix Eigenvalues of Triangular Matrices Associative Property of Matrix Multiplication |A| = |P||D||P-¹1 Determinant of a Matrix Product Definition of the Inverse of a Matrix Properties of the Identity Matrix Determinant of a Triangular Matrix Determinant of an Inverse Matrix Definition of a Diagonalizable Matrix Eigenvalues of Triangular Matrices Associative Property of Matrix Multiplication Similar Matrices Have the Same Eigenvalues |A| = |P||0|¹ = 101 Determinant of a Matrix Product Definition of the Inverse of a Matrix Properties of the Identity Matrix Determinant of a Triangular Matrix O Determinant of an Inverse Matrix Definition of a Diagonalizable Matrix Eigenvalues of Triangular Matrices Associative Property of Matrix Multiplication Similar Matrices Have the Same Eigenvalues |A|d₂d₂.dn O Determinant of a Matrix Product Definition of the Inverse of a Matrix Properties of the Identity Matrix Determinant of a Triangular Matrix Determinant of an Inverse Matrix Definition of a Diagonalizable Matrix Ⓒ Eigenvalues of Triangular Matrices Associative Property of Matrix Multiplication Prove that if matrix A is diagonalizable with in real eigenvalues A₁, A2, There exists an invertible matrix P and a diagonal matrix D, such that O Determinant of a Matrix Product Definition of the Inverse of a Matrix Properties of the Identity Matrix Determinant of a Triangular Matrix Determinant of an Inverse Matrix Definition of a Diagonalizable Matrix Eigenvalues of Triangular Matrices Associative Property of Matrix Multiplication Similar Matrices Have the Same Eigenvalues The eigenvalues of D are A₁, A₂, Determinant of a Matrix Product Definition of the Inverse of a Matrix Properties of the Identity Matrix Determinant of a Triangular Matrix Determinant of an Inverse Matrix Definition of a Diagonalizable Matrix Eigenvalues of Triangular Matrices Associative Property of Matrix Multiplication Similar Matrices Have the Same Eigenvalues The diagonal elements of D are A₁, A₂, A Determinant of a Matrix Product Definition of the Inverse of a Matrix Ⓒ Properties of the Identity Matrix Determinant of a Triangular Matrix O Determinant of an Inverse Matrix Definition of a Diagonalizable Matrix Eigenvalues of Triangular Matrices Athen |A|A₁A₂A. Complete the proof by justifying each step. P-1AP= D. |A| = |IAI Determinant of a Matrix Product Definition of the Inverse of a Matrix Properties of the Identity Matrix Determinant of a Triangular Matrix Determinant of an Inverse Matrix Definition of a Diagonalizable Matrix Eigenvalues of Triangular Matrices Associative Property of Matrix Multiplication Similar Matrices Have the Same Eigenvalues |A| = |(PP-1)A(PP-1)| Determinant of a Matrix Product Definition of the Inverse of a Matrix Properties of the Identity Matrix O Determinant of a Triangular Matrix Determinant of an Inverse Matrix Ⓒ Definition of a Diagonalizable Matrix Ⓒ Eigenvalues of Triangular Matrices Associative Property of Matrix Multiplication Similar Matrices Have the Same Eigenvalues |A| = |P(P-¹AP) P-1| = |PDP-11 Determinant of a Matrix Product Definition of the Inverse of a Matrix Properties of the Identity Matrix Determinant of a Triangular Matrix O Determinant of an Inverse Matrix Definition of a Diagonalizable Matrix Eigenvalues of Triangular Matrices Associative Property of Matrix Multiplication |A| = |P||D||P-¹1 Determinant of a Matrix Product Definition of the Inverse of a Matrix Properties of the Identity Matrix Determinant of a Triangular Matrix Determinant of an Inverse Matrix Definition of a Diagonalizable Matrix Eigenvalues of Triangular Matrices Associative Property of Matrix Multiplication Similar Matrices Have the Same Eigenvalues |A| = |P||0|¹ = 101 Determinant of a Matrix Product Definition of the Inverse of a Matrix Properties of the Identity Matrix Determinant of a Triangular Matrix O Determinant of an Inverse Matrix Definition of a Diagonalizable Matrix Eigenvalues of Triangular Matrices Associative Property of Matrix Multiplication Similar Matrices Have the Same Eigenvalues |A|d₂d₂.dn O Determinant of a Matrix Product Definition of the Inverse of a Matrix Properties of the Identity Matrix Determinant of a Triangular Matrix Determinant of an Inverse Matrix Definition of a Diagonalizable Matrix Ⓒ Eigenvalues of Triangular Matrices Associative Property of Matrix Multiplication

Answer rating: 100% (QA)

Solution To prove if matrix 4 is diagonalizable with n eigenv... View the full answer