Learn with Amar
Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Distinguish a minimal polynomial from a characteristic polynomial.
- Justify the conclusion "mA(t)=(t−2)(t−3)" using the stated assumptions.
Before you start
Matrix polynomials and diagonal matrices.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
For A=diag(2,2,3), find its minimal polynomial.
Why this math matters
Distinguish a minimal polynomial from a characteristic polynomial. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- Minimal polynomials are chosen monic.
- Matrices act on a finite-dimensional real vector space.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Find the shortest polynomial that annihilates a matrix
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For A=diag(2,2,3), find its minimal polynomial.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
χA(t)=(t−2)²(t−3)
The characteristic polynomial counts algebraic multiplicities.
Work through the mathematics
(A−2I)(A−3I)=0
On each diagonal entry, one of the factors vanishes.
Check the conclusion
mA(t)=(t−2)(t−3)
Both distinct eigenvalues must be roots of every annihilating polynomial, so degree two is minimal.
The result
mA(t)=(t−2)(t−3)
Both distinct eigenvalues must be roots of every annihilating polynomial, so degree two is minimal.
Common mistakes to catch
- The characteristic and minimal polynomials can have different degrees.
- Repeated eigenvalues alone do not determine Jordan block sizes.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is the minimal polynomial of 5I?
Show a hint
One linear factor annihilates it.
Reveal answer and explanation
t−5
A nonzero constant polynomial cannot annihilate a nonzero-dimensional identity matrix.
Practice 2
For [[2,1],[0,2]], is t−2 enough?
Show a hint
Subtract 2I.
Reveal answer and explanation
No; the minimal polynomial is (t−2)²
The first power leaves a nonzero nilpotent matrix, while its square is zero.
Take the idea with you
Reduce a high-degree matrix expression using its minimal polynomial.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Identify linear measurements as dual vectors
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