It is known that the third order matrix a satisfies the condition | e-A | = | 2e-a | = | 3e-a | to find the value of determinant | a |

It is known that the third order matrix a satisfies the condition | e-A | = | 2e-a | = | 3e-a | to find the value of determinant | a |

Let a's eigenvalues be x1, X2, X3, then e-A's eigenvalues are: 1-x1, 1-x2, 1-x32e-a's eigenvalues are: 2-x1, 2-x2, 2-x33e-a's eigenvalues are: 3-x1, 3-x2, 3-x3

Let a be a real symmetric matrix of order n with rank r, satisfying a ^ 4-3a ^ 3 + 3A ^ 2-2a = 0, then n eigenvalues of a?
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Let p be any eigenvalue of a, and a be the eigenvector of a belonging to P, then (a ^ 4-3a ^ 3 + 3A ^ 2-2a) a = (P ^ 4-3p ^ 3 + 3P ^ 2-2p) a = 0, that is, P (P-2) (P ^ 2-P + 1) = 0. Because the eigenvalue of a real symmetric matrix must be real, the eigenvalue of a can only be 0 or 2, and because it must be diagonalizable, the eigenvalue of a is 2 (2-fold), 0 (n-r-fold)

It is proved that a symmetric matrix with rank equal to R can be expressed as the sum of R symmetric matrices with rank equal to 1

Prompt, turn into contract standard type can