TIFR-2020-Maths-A: 17

Question

Which of the following statements is correct for every linear transformation $T:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ such that $T^{3}-T^{2}-T+I=0$?

• A. $T$ is invertible as well as diagonalizable.
• B. $T$ is invertible, but not necessearily diagonalizable.
• C. $T$ is diagonalizable, but not necessary invertible.
• D. None of the other three statements.

Answer 1

$\text{Every matrix  $A$ of size $m\times n$ can be represented as a linear transformation or a map from $\mathbb{R^n}\rightarrow \mathbb{R^m}$}$

$\text{So, here, think linear transformation T as a matrix A of size $3 \times 3$}$

$\text{It means, $A^3 – A^2 – A + I = 0$ }$

$A^2(A-I)-I(A-I) = 0$

$(A^2 – I)(A-I) = 0$

$\text{It means, Either $A^2 – I = 0$ or A-I=0}$

$\text{i.e. $A^2 = I$ or A=I}$

$\text{i.e. $A^{-1} = A$ or $A^{-1}=I$}$

$\text{It means, matrix $A$ is invertible.}$

$\text{matrix $A$ is not necessarily diagonalizable}$

$\text{Counterexample when $A^2 = I$:} $\begin{bmatrix} 1 &0 &0 \\ 0 &0 &1 \\ 0&1 &0 \end{bmatrix}