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Problem 120

a) Show that: (i) \(\mathrm{A} 0=0\) (ii) \(0 \mathrm{~A}=0\) (iii) \(\mathrm{AI}=\mathrm{A}\) (iv) \(\mathrm{IA}=\mathrm{A}\) where 0 and I denote the zero and identity matrices respectively, and \(\begin{array}{lll}\mathrm{A} & =12 & 1 & 3 \mid\end{array}\) \(\begin{array}{lll}14 & -1 & -1 \mid\end{array}\) b) Give examples of the following rules: (i) if A has a row of zeros, the same row of \(\mathrm{AB}\) consists of zeros, (ii) if \(\mathrm{B}\) has a column of zeros, the same column of AB consists of zeros.

Expert verified

In this exercise, we proved the following properties:
(i) A0=0, 0A=0
(ii) AI=A, IA=A
We also provided examples illustrating the two rules:
(i) If A has a row of zeros, the same row of AB consists of zeros.
Example: \(A = \begin{bmatrix} 1 & 2 \\ 0 & 0 \end{bmatrix}\), \(B = \begin{bmatrix} 3 & 4 \\ 5 & 6 \end{bmatrix}\), and \(AB = \begin{bmatrix} 13 & 16 \\ 0 & 0 \end{bmatrix}\).
(ii) If B has a column of zeros, the same column of AB consists of zeros.
Example: \(A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\), \(B = \begin{bmatrix} 5 & 0 \\ 6 & 0 \end{bmatrix}\), and \(AB = \begin{bmatrix} 17 & 0 \\ 39 & 0 \end{bmatrix}\).

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Chapter 4

Define (1) An upper triangular matrix. (2) A lower triangular matrix. (3) A properly triangular matrix. Give examples.

Chapter 4

Prove that if \(\mathrm{A}=\left(\mathrm{a}_{\mathrm{ij}}\right)\) and \(\mathrm{B}=\left(\mathrm{b}_{\mathrm{ij}}\right)\) are $\mathrm{m} \times \mathrm{n}$ matrices over a field \(\mathrm{K}\) and \(\mathrm{c}\) is an element of \(\mathrm{K}\), then \(\mathrm{A}+\mathrm{B}=\left(\mathrm{c}_{\mathrm{jj}}\right)\) where $c_{i j}=a_{i j}+b_{i j}, i=1,2, \ldots, m, j=1,2, \ldots, n\( and \)c A=\left(d_{i j}\right)$ where \(d_{i j}=c a_{i j}, i=1,2, \ldots, m, j=1,2, \ldots, n\)

Chapter 4

Find (i) \(A^{2}\) (ii) \(A^{3}\) (iii) \(A^{4}\) when $A=\begin{array}{rl}1 & 2 \mid \\ & \mid-1\end{array}$

Chapter 4

Let $\mathrm{A}=\begin{array}{cll}11 & 2 & 0 \mid \\ 13 & -1 & 4 \mid\end{array}\( Find (i) \)\mathrm{AA}^{\mathrm{t}}$, (ii) \(\mathrm{A}^{\mathrm{t}} \mathrm{A}\)

Chapter 4

Show that in matrix arithmetic we can have the following: (a) \(\quad \mathrm{AB} \neq \mathrm{BA}\). (b) \(\quad \mathrm{A} \neq 0, \mathrm{~B} \neq 0\) and yet, \(\mathrm{AB}=0\). (c) \(\quad A \neq 0\) and \(A^{2}=0\). (d) \(\quad \mathrm{A} \neq 0, \mathrm{~A}^{2} \neq 0\) and \(\mathrm{A}^{3}=0\). (e) \(\mathrm{A}^{2}=\mathrm{A}\) with \(\mathrm{A} \neq 0\) and $\mathrm{A} \neq \mathrm{I}$. (f) \(\quad A^{2}=\) I with \(A \neq I\) and \(A \neq-I\).

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