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GATE 2024 MA – Question 56

Linear Algebra · Eigenvalues and Canonical Forms · 2 marks · Multiple select

T:ℝ⁴→ℝ⁴ has only eigenvalues 1,2, dim ker(T−I)=1 and rank(T−2I)=2. Which Jordan forms are possible?

  1. $\begin{pmatrix}1&0&0&0\\0&2&0&0\\0&0&2&1\\0&0&0&2\end{pmatrix}$
  2. $\begin{pmatrix}1&0&0&0\\0&2&1&0\\0&0&2&1\\0&0&0&2\end{pmatrix}$
  3. $\begin{pmatrix}1&1&0&0\\0&1&0&0\\0&0&2&1\\0&0&0&2\end{pmatrix}$
  4. $\begin{pmatrix}1&1&0&0\\0&1&0&0\\0&0&2&0\\0&0&0&2\end{pmatrix}$

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Correct answer: (A) $\begin{pmatrix}1&0&0&0\\0&2&0&0\\0&0&2&1\\0&0&0&2\end{pmatrix}$; (D) $\begin{pmatrix}1&1&0&0\\0&1&0&0\\0&0&2&0\\0&0&0&2\end{pmatrix}$

Explanation

Nullity at eigenvalue 2 is 4−2=2, requiring two Jordan blocks for 2, while eigenvalue 1 requires one block. A has one size-one block for 1 and two blocks (sizes 1,2) for 2. D has one size-two block for 1 and two size-one blocks for 2. B,C each have only one block for 2 and fail its nullity. **A,D**.

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