Answer :
A 3 × 3 matrix does not have a characteristic polynomial. This means that the given matrix does not possess any eigenvalues.
The characteristic polynomial of a matrix can be found using either a cofactor expansion or the special formula for 3 × 3 determinants.
However, in this case, the question states that there is no characteristic polynomial for a 3 × 3 matrix.
A characteristic polynomial is a polynomial equation that helps find the eigenvalues of a matrix. It is determined by subtracting the identity matrix multiplied by the variable lambda from the given matrix, and then calculating its determinant.
Since the question states that there is no characteristic polynomial for a 3 × 3 matrix, we can conclude that the matrix in question does not have any eigenvalues.
Therefore, the answer is: There is no characteristic polynomial for the given 3 × 3 matrix.
The characteristic polynomial is determined by subtracting the identity matrix multiplied by lambda from the given matrix. In the case of a 3 × 3 matrix, this results in a 3 × 3 matrix with the variable lambda in the diagonal elements.
Taking the determinant of this matrix would give us a polynomial equation, which represents the characteristic polynomial.
However, the question specifically states that there is no characteristic polynomial for a 3 × 3 matrix. This implies that the given matrix does not have any eigenvalues.
In summary, a 3 × 3 matrix does not have a characteristic polynomial. This means that the given matrix does not possess any eigenvalues.
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