If \(1\) is the only eigenvalue of \(\mathbf{A},\) then \(\mathbf{A}=\mathbf{1}_{n}.\)

• True
• False

The eigenvalues of an anti-symmetric matrix \(\mathbf{A}\in M_{2,2}(\mathbb{C})\) with real entries are pure imaginary.

• True
• False

\(1\) is an eigenvalue of \(f:\mathsf P_2(\mathbb{R})\to \mathsf P_2(\mathbb{R}),\)\(p\mapsto f(p) = p+\frac{\mathrm d}{\mathrm dx}p\)

• True
• False

The algebraic multiplicity of the eigenvalue \(1\) of \(f:\mathsf P_2(\mathbb{R})\to \mathsf P_2(\mathbb{R}),\)\(p\mapsto f(p) = p+\frac{\mathrm d}{\mathrm dx}p\) equals its geometric multiplicity.

• True
• False

Let \(f:\mathsf P_2(\mathbb{R})\to \mathsf P_2(\mathbb{R}),\)\(p\mapsto f(p) = p+\frac{\mathrm d}{\mathrm dx}p.\) Then \(\dim(\operatorname{Eig}_f(1))=1.\)

• True
• False

Let \(f:\mathbb{K}^n\to\mathbb{K}^n\) be an endomorphism. If \(\vec v,\vec w\in \mathbb{K}^n\) are eigenvectors of \(f\) with respect to \(\lambda,\mu\in\mathbb{K}\) respectively, where \(\lambda\ne \mu,\) then \(\vec v\) and \(\vec w\) are linearly independent.

• True
• False

If \(\lambda\ne\mu\) are eigenvalues of \(f:\mathbb{R}^3\to\mathbb{R}^3,\) then \(\operatorname{Eig}_f(\lambda)\oplus \operatorname{Eig}_f(\mu)=\mathbb{R}^3.\)

• True
• False

If \(\lambda\ne\mu\) are eigenvalues of \(f:\mathbb{R}^2\to\mathbb{R}^2,\) then \(\operatorname{Eig}_f(\lambda)\oplus \operatorname{Eig}_f(\mu)=\mathbb{R}^2.\)

• True
• False

Let \(f:V\to V\) be an endomorphism, where \(V\) is a \(\mathbb{C}\)-vector space. If all eigenvalues of \(f\) are elements of \(\mathbb{C}\setminus\{0\},\) then \(\det(f)\ne 0.\)

• True
• False

Let \(f:V\to V\) be an endomorphism, where \(V\) is a \(\mathbb{C}\)-vector space. If \(\lambda\in\mathbb{C}\) is an eigenvalue of \(f,\) then so is \(\bar\lambda.\)

• True
• False

Let \(\mathbf{A}\in M_{n,n}(\mathbb{C})\) be a matrix with real entries. If \(\lambda\) is an eigenvalue of \(\mathbf{A},\) then so is \(\bar\lambda.\)

• True
• False

Rotations in \(\mathbb{R}^2\) have zero eigenvalues.

• True
• False

Reflections in \(\mathbb{R}^2\) have two eigenvalues.

• True
• False

The linear endomorphism corresponding to \(\mathbf{A}\in M_{2,2}(\mathbb{R})\) can have either \(0,\) \(1,\) or \(2\) distinct eigenvalues.

• True
• False

The linear endomorphism corresponding to \(\mathbf{A}\in M_{2,2}(\mathbb{C})\) can have either \(0,\) \(1,\) or \(2\) distinct eigenvalues.

• True
• False