Matrices & Determinants
Matrices and Determinants
Allen Star Batch
Grade 12
Question:
Which of the following matrices do not have eigen values 1 and $-1$
$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
$\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$
$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$
Step-by-Step Solution
Key Concept: The eigenvalues of a matrix A are found by solving det(A - λI) = 0. For the identity matrix I, all eigenvalues equal 1 with multiplicity equal to the matrix dimension, since det(I - λI) = (1-λ)^n = 0 yields only λ = 1.
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