The matrixC=⎡⎣⎢（−6，2，-6），（-5，1，-6），（4，-1，5）⎤⎦⎥has two distinct eigenvalues such that λ1<λ2.The smaller eigenvalue λ1= has algebraic multiplicity and geometric multiplicity .The larger eigenvalue λ2= has algebraic multiplicity and geometric multiplicity .Is the matrix C diagonalisable? choose
Question
The matrixC=⎡⎣⎢（−6，2，-6），（-5，1，-6），（4，-1，5）⎤⎦⎥has two distinct eigenvalues such that λ1<λ2.The smaller eigenvalue λ1= has algebraic multiplicity and geometric multiplicity .The larger eigenvalue λ2= has algebraic multiplicity and geometric multiplicity .Is the matrix C diagonalisable? choose
Solution
To answer this question, we first need to find the eigenvalues of the matrix C. The eigenvalues are the roots of the characteristic polynomial, which is found by subtracting λ from the diagonal entries of the matrix and then taking the determinant.
The matrix C is:
C = [ -6 2 -6 -5 1 -6 4 -1 5]
The characteristic polynomial is found by calculating the determinant of (C - λI), where I is the identity matrix and λ is a scalar. This gives us:
det(C - λI) = det([ -6-λ 2 -6 -5 1-λ -6 4 -1 5-λ])
Solving this equation will give us the eigenvalues of the matrix.
The algebraic multiplicity of an eigenvalue is the number of times it appears as a root of the characteristic polynomial. The geometric multiplicity of an eigenvalue is the dimension of the associated eigenspace.
A matrix is diagonalizable if and only if the algebraic multiplicity of each of its eigenvalues equals its geometric multiplicity.
Without the specific values of the eigenvalues and their multiplicities, we cannot determine if the matrix C is diagonalizable.
Similar Questions
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The characteristic polynomial of a 3×33×3 matrix can be factorised as (2−λ)2(5−λ)(2−𝜆)2(5−𝜆). Which of the following statements are true? Select all that apply.The geometric multiplicity of the eigenvalue λ=𝜆= 55 could be 22The number of linearly independent eigenvectors for the eigenvalue λ=𝜆= 55 must be 11The geometric multiplicity of the matrix cannot be greater than 22The geometric multiplicity of the eigenvalue λ=𝜆= 22 could be 22The algebraic multiplicity of the matrix is 22The algebraic multiplicity of the eigenvalue λ=𝜆= 22 cannot be determined from the information we are givenThe eigenvalue λ=𝜆= 22 could have infinitely many eigenvector directionsThe algebraic multiplicity of the eigenvalue λ=𝜆= 22 is 2
diagonalizing a matrix;
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