Question

Prove that if matrix A is diagonalizable with n real eigenvalues λι, λ2-..,An, then AI-λιλ2" λπ. Complete the proof by justifying each step. There exists an invertible matrix P and a diagonal matrix D, such that P1AP -D. -JIAT O Determinant of a Matrix Product O Definition of the Inverse of a Matrix O Properties of the Identity Matrix O Determinant of a Triangular Matrix O Determinant of an Inverse Matrix O Definition of a Diagonalizable Matrix O Eigenvalues of Triangular Matrices O Associative Property of Matrix Multiplication O Determinant of a Matrix Product O Definition of the Inverse of a Matrix O Properties of the Identity Matrix O Determinant of a Triangular Matrix O Determinant of an Inverse Matrix O Determinant of a Matrix Product O Definition of the Inverse of a Matrix O Properties of the Identity Matrix Determinant of a Triangular Matrix O Determinant of an Inverse Matrix Definition of a Diagonalizable Matrisx Definition of a Diagonalizable Matrix O Eigenvalues of Triangular Matrices O Associative Property of Matrix Multiplication O Similar Matrices Have the Same Eigenvalues Eigenvalues of Triangular Matrices O Associative Property of Matrix Multiplication O Similar Matrices Have the Same Eigenvalues Similar Matrices Have the Same Eigenvalues The eigenvalues of D are λ, λ2, - ..,An. AlIPPA(PP-1)I IPI O Determinant of a Matrix Product O Definition of the Inverse ofa Matrix O Properties of the Identity Matrix O Determinant of a Triangular Matrix O Determinant of an Inverse Matrix O Determinant of a Matrix Product O Definition of the Inverse of a Matrix O Properties of the Identity Matrix O Determinant of a Triangular Matrix O Determinant of an Inverse Matrix O Definition of a Diagonali O Eigenvalues of Triangular Matrices O Determinant of a Matrix Product O Definition of the Inverse of a Matrix O Properties of the Identity Matrix O Determinant of a Triangular Matrix O Determinant of an Inverse Matrix O Definition of a Diagonalizable Matrix O Eigenvalues of Triangular Matrices O Associative Property of Matrix Multiplication O Similar Matrices Have the Same Eigenvalues Definition of a Diagonalizable Matrix Eigenvalues of Triangular Matrices Associative Property of Matrix Multiplication zable Matrix Associative Property of Matrix Multiplication O Similar Matrices Have the Same Eigenvalues O Similar Matrices Have the Same Eigenvalues The diagonal elements of D are λνλ2, AI , O Determinant of a Matrix Product O Definition of the Inverse ofa Matrix O Properties of the Identity Matrix O Determinant of a Triangular Matrix O Determinant of an Inverse Matrix O Definition of a Diagonalizable Matrix O Eigenvalues of Triangular Matrices Determinant of a Matrix Product O Definition of the Inverse ofa Matrix Properties of the Identity Matrix O Determinant of a Triangular Matrix O Determinant of an Inverse Matrix O Determinant of a Matrix Product O Definition of the Inverse of a Matrix O Properties of the Identity Matrix Determinant of a Triangular Matrix O Determinant of an Inverse Matrix O Definition of a Diagonalizable Matrix O Eigenvalues of Triangular Matrices O Associative Property of Matrix Multiplication O Definition of a D agonalizable Matrix O Eigenvalues of Triangular Matrices Associative Property of Matrix Multiplication

Answer 1

Definition of diagonalizable matrix properties of identity matrix Determinant of matrix product similar matrices have same eigen values definition of the inverse of a matrix determinant of an Inverse matrix Eigen values of triangular matrices associative properties of matrix multiplication determinant of triangular matrix