If \(\mathbf A,\mathbf B\in M_{n,n}(\mathbb K),\) then \(\mathbf A\mathbf B = \mathbf{0}_n\) implies \(\mathbf A=\mathbf{0}_n\) or \(\mathbf B = \mathbf{0}_n.\)

• True
• False

If \(\mathbf A,\mathbf B\in M_{n,n}(\mathbb K)\) are such that \(\mathbf{A}\mathbf{B}=\mathbf{0}_n,\) then \(\mathbf{B}\mathbf{A}=\mathbf{0}_n.\)

• True
• False

If \(\mathbf D_1,\mathbf D_2\in M_{n,n}(\mathbb{K})\) are diagonal matrices, then \(\mathbf D_1\mathbf D_2=\mathbf D_2\mathbf D_1.\)

• True
• False

If \(\mathbf{A},\mathbf{B}\) are matrices such that \(\mathbf{A}\mathbf{B}\) and \(\mathbf{B}\mathbf{A}\) are defined, then \(\mathbf{A}\mathbf{B}\) and \(\mathbf{B}\mathbf{A}\) are of the same size.

• True
• False

If \(\mathbf{A},\mathbf{B}\) are matrices such that \(\mathbf{A}\mathbf{B}\) and \(\mathbf{B}\mathbf{A}\) are defined and \(\mathbf{A}\mathbf{B}\) and \(\mathbf{B}\mathbf{A}\) are of the same size, then \(\mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A}.\)

• True
• False

If \(\mathbf{A},\mathbf{B}\in M_{n,n}(\mathbb{K})\) are such that \(\mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A},\) then \(\mathbf{A}\mathbf{B}\) must be symmetric.

• True
• False

If, \(\mathbf{A}\in M_{m,r}(\mathbb{K}),\) \(\mathbf{B}\in M_{r,n}(\mathbb{K})\) and \(\mathbf{C}\in M_{r,m}(\mathbb{K}),\) then the products \(\mathbf{A}\mathbf{B},\) \(\mathbf{A}\mathbf{C},\) \(\mathbf{C}\mathbf{A}\) and \(\mathbf{C}^T\mathbf{B}\) are all defined.

• True
• False

Every \(1\)-homogeneous map \(f:\mathbb{R}\to\mathbb{R}\) is additive.

• True
• False

If \(\mathbf A\in M_{2,2}(\mathbb{R})\) is such that \(\mathbf A^2=\mathbf{0}_2,\) then \(\mathbf A=\mathbf{0}_2.\)

• True
• False

Given \(\mathbf{A}\in M_{n,n}(\mathbb{K}),\) the matrix \(\mathbf{A}\mathbf{A}^T\) is always square.

• True
• False

Given \(\mathbf{A}\in M_{n,n}(\mathbb{K}),\) the matrix \(\mathbf{A}\mathbf{A}^T\) is always symmetric.

• True
• False

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

• True
• False

If \(\mathbf{A},\mathbf{B}, \mathbf{C}\in M_{n,n}(\mathbb K),\) and \(\mathbf{A}\mathbf{B}=\mathbf{A}\mathbf{C},\) then \(\mathbf{B}=\mathbf{C}.\)

• True
• False

Given \(\mathbf{A}\in M_{n,n}(\mathbb{K}),\) the matrix \(\mathbf{A}+\mathbf{A}^T\) is always symmetric.

• True
• False

Given \(\mathbf{A}\in M_{n,n}(\mathbb{K}),\) the matrix \(\mathbf{A}-\mathbf{A}^T\) is always anti-symmetric.

• True
• False