Question
Write down examples of (a) a $3 \times 3$ matrix; (b) a $2 \times 3$ matrix; (c) a matrix with 3 rows and 4 columns; $(d)$ a row vector with 4 entries; $(e)$ a column vector with 3 entries; (f) a matrix that is both a row vector and a column vector.
Step 1
A $3 \times 3$ matrix has 3 rows and 3 columns. We can fill it with any numbers. For example: \[ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{array} \] Show more…
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Key Concepts
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Give an example of a $3 \times 3$ matrix $A$ with all nonzero entries such that det $A=13$
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a. Give an example of a $3 \times 3$ matrix $A$ with as many nonzero entries as possible such that both $\operatorname{span}\left(\vec{e}_{1}\right)$ and $\operatorname{span}\left(\vec{e}_{1}, \vec{e}_{2}\right)$ are $A$ -invariant subspaces of $\mathbb{R}^{3}$ See Exercise 65 b. Consider the linear space $V$ of all $3 \times 3$ matrices $A$ such that both $\operatorname{span}\left(\vec{e}_{1}\right)$ and $\operatorname{span}\left(\vec{e}_{1}, \vec{e}_{2}\right)$ are $A$ -invariant subspaces of $\mathbb{R}^{3}$. Describe the space $V$ (the matrices in $V$ "have a name"), and determine the dimension of $V$
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Diagonalization
Consider the matrix $\left[\begin{array}{rrr}{-2} & {3} & {1} \\ {0} & {5} & {-3} \\ {1} & {4} & {8}\end{array}\right]$ and answer the following. (a) What are the elements of the second row? (b) What are the elements of the third column? (c) Is this a square matrix? Explain why or why not. (d) Give the matrix obtained by interchanging the first and third rows. (e) Give the matrix obtained by multiplying the first row by $-\frac{1}{2}$ (f) Give the matrix obtained by multiplying the third row by 3 and adding to the first row.
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