GATE 2026 MN – Question 11

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

For $P=\begin{pmatrix}2&1\\0&3\end{pmatrix}$ and identity I, which polynomial identity follows from Cayley–Hamilton?

$P^2-5P+6I=0$
$P^2-5P+5I=0$
$P^2-6P+5I=0$
$P^2-4P+3I=0$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) $P^2-5P+6I=0$

Explanation

The triangular matrix has eigenvalues2 and3, so its characteristic polynomial is (λ−2)(λ−3)=λ²−5λ+6. Cayley–Hamilton substitutes the matrix into its own characteristic polynomial, giving P²−5P+6I=0. Thus A.

More Linear Algebra questions

GATE 2026 MN Q38 – Let x=(1,2,1)T, b=(-16,-3,7)T, and P-1=pmatrixe&-6&-7\&9&11\\-1&-2&gpmatrix. Given Px=b…
GATE 2025 MN Q14 – A3×3 matrix has rank2. Find its determinant.
GATE 2025 MN Q55 – The characteristic polynomial is x3-6x2+11x-6. One eigenvalue is1. Find the sum of the…
GATE 2024 MN Q38 – Given A=pmatrix1&4&3\\5&2&1\\6&4&3pmatrix, classify B=A-AT.
GATE 2023 MN Q27 – Matrix A is negative definite. Which statement is NOT correct? Use the conventional real…
GATE 2023 MN Q41 – Three straight-line equations have (slope, intercept) pairs (2, 3), (4, 5), and (6, 2).…

All Linear Algebra questions →