20.7: End-of-Appendix Materials
- Page ID
- 57815
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Exercises
- Prove \(\mathbf{A} + \mathbf{0} = \mathbf{0} + \mathbf{A} = \mathbf{A}\).
- Prove matrix addition is commutative.
- Prove matrix addition is associative.
- Prove that scalar multiplication is associative.
- Prove that scalar multiplication is distributive over addition.
- Using a counterexample, prove that matrix multiplication is not commutative when only one of the matrices is diagonal (thus showing that \emph{both} must be diagonal).
- Let \(\mathbf{A}\) be any square matrix. Show that \(\frac{(\mathbf{A}+\mathbf{A}^\prime)}{2}\) is symmetric.
- Determine the determinant of \(\mathbf{J}_{10}\).
- Determine the rank of \(\mathbf{J}_{10}\).
- Prove that \(\mathbf{j}_{10}\ \mathbf{j}_{10}^\prime\) is not positive definite.
- Prove that \(\mathbf{j}_{10}^\prime\ \mathbf{j}_{10}\) \emph{is} positive definite.
- Prove \((\mathbf{AB})^{\prime} = \mathbf{B^{\prime}A^{\prime}}\) from Section 20.5.
- Prove \((\mathbf{AB})^{-1} = \mathbf{B^{-1}A^{-1}}\) from Section 20.5.
- Prove Lemma 20.5.1.
- Prove Lemma 20.5.2.
- Prove Lemma 20.5.3.
- Prove Lemma 20.6.5.
- Prove Lemma 20.6.6.
- Prove Lemma 20.6.7.