Let \( T : {R}^4 \to {R}^4 \) be an \( {R} \)-linear transformation such that 1 and 2 are the only eigenvalues of \( T \). Suppose the dimensions of \({Kernel}(T - I_4)\) and \({Range}(T - 2I_4)\) are 1 and 2, respectively. Which of the following is/are possible (upper triangular) Jordan canonical form(s) of \( T \)?
Show Hint
Solution and Explanation
Step 1: Eigenvalue properties and Jordan blocks. The eigenvalues of \( T \) are 1 and 2. The dimensions of the kernel and range provide information about the size of the Jordan blocks. - Dimension of \({Kernel}(T - I_4) = 1\): Indicates one Jordan block corresponding to \( \lambda = 1 \). - Dimension of \({Range}(T - 2I_4) = 2\): Indicates two Jordan blocks for \( \lambda = 2 \).
Step 2: Analyzing each option. - (1): This corresponds to one Jordan block for \( \lambda = 1 \) and two for \( \lambda = 2 \), consistent with the given conditions. - (2): This has more than one Jordan block for \( \lambda = 1 \), violating the kernel dimension condition. - (3): This includes a defective Jordan block for \( \lambda = 1 \), which is inconsistent with the kernel dimension condition. - (4): This corresponds to one Jordan block for \( \lambda = 1 \) and two for \( \lambda = 2 \), consistent with the conditions.
Step 3: Conclusion. The possible Jordan canonical forms are \( {(1) and (4)} \).
Top Questions on Product of Matrices
- \(\text{ If } P = \begin{bmatrix} -1 \\ 2 \\ 1 \end{bmatrix} \text{ and } Q = \begin{bmatrix} 2 & -4 & 1 \end{bmatrix} \text{ are two matrices, then } (PQ)^T \text{ will be:}\)
- CUET (UG) - 2024
- Mathematics
- Product of Matrices
Consider the following Linear Programming Problem $ P $: Minimize $ x_1 + 2x_2 $, subject to
$ 2x_1 + x_2 \leq 2 $,
$ x_1 + x_2 = 1 $,
$ x_1, x_2 \geq 0 $.
The optimal value of the problem $ P $ is equal to:- GATE MA - 2024
- Mathematics
- Product of Matrices
Let $D = \{(x, y) \in \mathbb{R}^2 : x > 0 \text{ and } y > 0\}$. If the following second-order linear partial differential equation
$y^2 \frac{\partial^2 u}{\partial x^2} - x^2 \frac{\partial^2 u}{\partial y^2} + y \frac{\partial u}{\partial y} = 0$ on $D$
is transformed to
$\left( \frac{\partial^2 u}{\partial \eta^2} - \frac{\partial^2 u}{\partial \xi^2} \right) + \left( \frac{\partial u}{\partial \eta} + \frac{\partial u}{\partial \xi} \right) \frac{1}{2\eta} + \left( \frac{\partial u}{\partial \eta} - \frac{\partial u}{\partial \xi} \right) \frac{1}{2\xi} = 0$ on $D$,
for some $a, b \in \mathbb{R}$, via the coordinate transform $\eta = \frac{x^2}{2}$ and $\xi = \frac{y^2}{2}$, then which one of the following is correct?- GATE MA - 2024
- Mathematics
- Product of Matrices
- Let \( T, S : {R}^4 \to {R}^4 \) be two non-zero, non-identity \( {R} \)-linear transformations. Assume \( T^2 = T \). Which of the following is/are true?
- GATE MA - 2024
- Mathematics
- Product of Matrices
Let \( p_1<p_2 \) be the two fixed points of the function \( g(x) = e^x - 2 \), where \( x \in {R} \). For \( x_0 \in {R} \), let the sequence \( (x_n)_{n \geq 1} \) be generated by the fixed-point iteration \[ x_n = g(x_{n-1}), \quad n \geq 1. \] Which one of the following is/are correct?
- GATE MA - 2024
- Mathematics
- Product of Matrices
Questions Asked in GATE MA exam
- Let \( \alpha, \beta \) be distinct non-zero real numbers, and let \( Q(z) \) be a polynomial of degree less than 5. If the function \[ f(z) = \frac{\alpha^6 \sin \beta z - \beta^6 (e^{2az} - Q(z))}{z^6} \] satisfies Morera's theorem in \( \mathbb{C} \setminus \{0\} \), then the value of \( \frac{\alpha}{4\beta} \) is equal to (in integer).
- GATE MA - 2025
- Function
- A shop has 4 distinct flavors of ice-cream. One can purchase any number of scoops of any flavor. The order in which the scoops are purchased is inconsequential. If one wants to purchase 3 scoops of ice-cream, in how many ways can one make that purchase?
- GATE MA - 2025
- GATE AG - 2025
- Combinations
A square paper, shown in figure (I), is folded along the dotted lines as shown in figures (II) and (III). Then a few cuts are made as shown in figure (IV). Which one of the following patterns will be obtained when the paper is unfolded?
- GATE MA - 2025
- GATE AG - 2025
- Paper Folding and Cutting
- A fair six-faced dice, with the faces labelled ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, and ‘6’, is rolled thrice. What is the probability of rolling ‘6’ exactly once?
- GATE MA - 2025
- GATE AG - 2025
- Probability
In the diagram, the lines QR and ST are parallel to each other. The shortest distance between these two lines is half the shortest distance between the point P and the line QR. What is the ratio of the area of the triangle PST to the area of the trapezium SQRT?
Note: The figure shown is representative