Eigenvalues and eigenvectors: not losing marks in the determinant step
Ask a B.Tech student to explain what an eigenvalue means and most can give a reasonable answer. Ask them to find one in an exam and the mark often disappears in the second line — not because the method was wrong, but because a 3×3 determinant was expanded carelessly under time pressure. Four checks fix most of it.
- Expand along the row with the most zeros.
Students default to the first row out of habit, even when the second row has a zero in it. A single zero removes an entire 2×2 determinant from the working. Choosing the easiest row is worth doing consciously.
- Write the full characteristic polynomial before you touch it.
Rushing into factorising a half-simplified expression is where sign errors creep in. Write the full cubic out first — every term, every sign — then start factorising.
- Check an eigenvalue before finding its eigenvector.
Substitute each root back into the characteristic equation. If it does not give zero, the determinant was expanded wrongly. Finding that out in thirty seconds beats spending four minutes on an eigenvector for a value that was never correct.
- Watch the eigenvector step’s own trap.
Once (A − λI)v = 0 is set up, students often solve for one variable and assume the rest follow. Write out every equation from the matrix, not just the first, and check your final vector satisfies all of them.
None of this is new mathematics — it is exam discipline layered on top of what you already know. If this is where your marks are disappearing, bring your own linear algebra paper in to Sector 12 and we will go through it line by line.
