One complete 2x2 Characteristic Polynomial calculation
For [[4,2],[1,3]], trace is 7 and determinant is 10, giving λ²−7λ+10.
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2x2 Characteristic Polynomial Calculator
Form λ²−tr(A)λ+det(A) for a 2×2 matrix. A checkable formula accompanies characteristic polynomial instead of leaving an unexplained number.
2x2 Characteristic Polynomial inputs
Known quantities
How 2x2 Characteristic Polynomial works
The characteristic polynomial det(λI−A) has roots equal to the matrix eigenvalues.
Conditions that affect 2x2 Characteristic Polynomial
Compute trace and determinant, place them into the monic quadratic, and check the constant term. 2x2 Characteristic Polynomial also connects to linear coefficient.
Where 2x2 Characteristic Polynomial applies
It connects trace, determinant, eigenvalues, stability, and repeated transformations.
Sign conventions differ between det(λI−A) and det(A−λI) by a predictable factor; this page uses the monic form. If the 2x2 Characteristic Polynomial assumptions do not fit, consider polynomial roots.
Checking both coefficients
The 2x2 Characteristic Polynomial setup records 2×2 matrix A. For a 2×2 matrix, the linear coefficient is the negative trace and the constant term is the determinant.
For 2x2 Characteristic Polynomial, attach the coefficient convention and coordinate basis to the saved array.
Boundary checks for 2x2 Characteristic Polynomial
Keep the 2x2 Characteristic Polynomial row or coordinate order for 2×2 matrix A. Read the result in the same 2x2 Characteristic Polynomial order.
Check one 2x2 Characteristic Polynomial component by hand. Substitute or multiply the result back to verify 2x2 Characteristic Polynomial.
The relationship between 2×2 matrix A and Characteristic polynomial provides another check on 2x2 Characteristic Polynomial. Move 2×2 matrix A slightly while keeping the stated condition constant, then decide in advance whether Characteristic polynomial ought to rise, fall, or remain unchanged.
Trace 2x2 Characteristic Polynomial back through 2×2 matrix A and the fixed condition. Those entries should support the displayed Characteristic polynomial. For a sensitivity check, alter the fixed condition only. Compare that 2x2 Characteristic Polynomial result with the first Characteristic polynomial, keeping both cases visible.
Questions about 2x2 Characteristic Polynomial
What does 2x2 Characteristic Polynomial calculate?
The characteristic polynomial det(λI−A) has roots equal to the matrix eigenvalues.
When is 2x2 Characteristic Polynomial useful?
It connects trace, determinant, eigenvalues, stability, and repeated transformations.
What can make 2x2 Characteristic Polynomial misleading?
Sign conventions differ between det(λI−A) and det(A−λI) by a predictable factor; this page uses the monic form.