Matrices & Determinants
Matrices and Determinants
star_batch_jee_advanced_2025
Grade None
Question:
Which of the following matrices do not have eigen values 1 and $-1$
1. \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}
2. \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}
3. \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
4. \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}
Step-by-Step Solution
Key Concept: The identity matrix has all eigenvalues equal to 1, each with multiplicity equal to the matrix dimension.
The matrix $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ has eigenvalues 1 and 1 (both repeated), since it is the identity matrix and $\det(I - \lambda I) = (1-\lambda)^2 = 0$ gives $\lambda = 1$ with multiplicity 2.
Correct Answer: 3