If \(\mathbf{A}\in M_{n,n}(\mathbb{K})\) has fewer than \(n\) distinct eigenvalues, \(\mathbf{A}\) cannot be diagonalisable.

• True
• False

If \(\mathbf{A}\in M_{n,n}(\mathbb{K})\) is diagonalisable and invertible, so is \(\mathbf{A}^{-1}.\)

• True
• False

If \(\mathbf{A}\in M_{n,n}(\mathbb{K})\) is diagonalisable, so is \(\mathbf{A}^T.\)

• True
• False

If \(\mathbf{A}\in M_{n,n}(\mathbb{K}),\) \(\mathbf{A}\) and \(\mathbf{A}^T\) have the same eigenvalues.

• True
• False

If \(\mathbf{A}\in M_{n,n}(\mathbb{K}),\) \(\mathbf{A}\) and \(\mathbf{A}^T\) have the same eigenvectors.

• True
• False

If \(\mathbf{A}\in M_{2,2}(\mathbb{R})\) is such that \(\operatorname{char}_\mathbf{A}(x) = x^2+bx+c\) with \(b^2-4c>0,\) then \(\mathbf{A}\) is diagonalisable.

• True
• False

The map \(M_{n,n}(\mathbb{R})\to \mathsf P_n(\mathbb{R}), \mathbf{A}\to\operatorname{char}_\mathbf{A}\) is linear.

• True
• False

Let \(\mathbf{A},\mathbf{B}\in M_{n,n}(\mathbb{R})\) and let \(\mathbf{C}\in M_{n,n}(\mathbb{R})\) be invertible such that \(\mathbf{C}^{-1}\mathbf{A}\mathbf{C}\) and \(\mathbf{C}^{-1}\mathbf{B}\mathbf{C}\) are diagonal. Then \(\mathbf{A}\mathbf{B}= \mathbf{B}\mathbf{A}.\)

• True
• False

Given \(\mathbf{A}\in M_{n,n}(\mathbb{R}),\) with spectrum \(\{\lambda_1,\dots,\lambda_n\},\) then \(\operatorname{Tr}(\mathbf{A})=\sum_{i=1}^n \lambda_i.\)

• True
• False

Given \(\mathbf{A}\in M_{n,n}(\mathbb{R}),\) with spectrum \(\{\lambda_1,\dots,\lambda_n\},\) then \(\det(\mathbf{A})=\prod_{i=1}^n \lambda_i.\)

• True
• False

An endomorphism \(f:V\to V\) with spectrum \(\{-1\}\) must be an involution.

• True
• False

Given an invertible matrix \(\mathbf{A}\in M_{n,n}(\mathbb{K}),\) \(\mathbf{A}\) and \(\mathbf{A}^{-1}\) have the same eigenvalues.

• True
• False

Let \(\mathbf{A}\in M_{n,n}(\mathbb{K})\) be invertible. If \(\mathbf{A}\) and \(\mathbf{A}^{-1}\) have the same eigenvalues, then \(\mathbf{A}=\mathbf{1}_{n}.\)

• True
• False

Let \(\mathbf{A}\in M_{n,n}(\mathbb{K})\) be invertible. If \(\mathbf{A}\) and \(\mathbf{A}^{-1}\) have the same positive eigenvalues, then \(\mathbf{A}=\mathbf{1}_{n}.\)

• True
• False

Let \(f:V\to V\) be an endomorphism of the \(\mathbb{K}\)-vector space \(V\) and let \(U \subset V\) be a subspace which is stable under \(f.\) Then \(U\) is an eigenspace of \(f.\)

• True
• False

Let \(f:V\to V\) be an endomorphism of the \(\mathbb{K}\)-vector space \(V\) and let \(U \subset V\) be a subspace which is stable under \(f.\) Then \(U\) must be an eigenspace of \(f.\)

• True
• False

Let \(f:V\to V\) be an endomorphism of the \(\mathbb{K}\)-vector space \(V\) such that \(\lambda\ne\mu\) are eigenvalues of \(f.\) Then the subspace \(U=\operatorname{Eig}_f(\lambda)\oplus\operatorname{Eig}_f(\mu)\) is stable under \(f.\)

• True
• False