If \(\sigma\) is a permutation, then \(\mathbf P_\sigma^{-1}=\mathbf P_{\sigma}.\)

• True
• False

If \(\sigma\in S_n\) then \(\sigma^n = 1.\)

• True
• False

If \(\sigma,\tau\in S_n\) then \(\sigma^n = \tau^n\) implies \(\sigma=\tau.\)

• True
• False

If \(\sigma,\tau\in S_n\) then \(\sigma\circ\tau\) implies \(\tau\circ\sigma.\)

• True
• False

It holds that \(|\det(\mathbf{A})|=\sqrt{\det(\mathbf{A}^T\mathbf{A})}\) for all \(\mathbf{A}\in M_{n,n}(\mathbb{K}).\)

• True
• False

Let \(\mathbf{A}\in M_{2n+1,2n+1}(\mathbb{K})\) be anti-symmetric. Then \(\det(\mathbf{A})=0.\)

• True
• False

No anti-symmetric matrix \(\mathbf{A}\) is invertible.

• True
• False

If \(\mathbf{A}\in M_{n,n}(\mathbb{K})\) is such that \(\mathbf{A}^T\mathbf{A}=\mathbf{1}_{n},\) then \(\det(\mathbf{A})=1.\)

• True
• False

A matrix \(\mathbf{A}\in M_{n,n}(\mathbb{K})\) is invertible if and only if \(\operatorname{Adj}(\mathbf{A})\) is.

• True
• False

Given \(\mathbf{A},\mathbf{B}\in M_{n,n}(\mathbb{K}),\) then \(\det(\mathbf{A})\det(\mathbf{B})=\det(\mathbf{B})\det(\mathbf{A}).\)

• True
• False

Given \(\mathbf{A},\mathbf{B}\in M_{n,n}(\mathbb{K}),\) then \(\det(\mathbf{A}\mathbf{B})=\det(\mathbf{A})+\det(\mathbf{B}).\)

• True
• False

If \(\mathbf{A},\mathbf{B}\in M_{n,n}(\mathbb{K})\) are such that \(\det(\mathbf{A})=\det(\mathbf{B}),\) then there exists an invertible matrix \(\mathbf{C}\in M_{n,n}(\mathbb{K})\) such that \(\mathbf{B}= \mathbf{C}\mathbf{A}\mathbf{C}^{-1}.\)

• True
• False

A square matrix is non-invertible if and only if its transpose is non-invertible.

• True
• False

The matrix \(\begin{psmallmatrix}a & b\\ 5 & b\end{psmallmatrix}\) is invertible if and only if \(a\ne5\) and \(b\ne0.\)

• True
• False

The matrix \(\begin{psmallmatrix}x+\mathrm i & 0\\ 0 & x-\mathrm i\end{psmallmatrix}\) is invertible for all \(x\in\mathbb{R}.\)

• True
• False

If \(\mathbf{A}\in M_{m,n}(\mathbb{K}),\) where \(m\ne n,\) then \(\det(\mathbf{A}^T\mathbf{A})=\det(\mathbf{A}\mathbf{A}^T).\)

• True
• False

It holds that \(\operatorname{Adj}(\mathbf{A}^T)=\operatorname{Adj}(\mathbf{A})^T\) for all \(\mathbf{A}\in M_{n,n}(\mathbb{K}).\)

• True
• False