Using Matrices to Represent, Manipulate Data - Precalculus

1. Identify whether $AB$ is defined if $A$ is $2\times 3$ and $B$ is $2\times 2$.

   Not defined, since $3\neq 2$ for the inner dimensions. Column count of $A$ doesn't match row count of $B$.

2. Compute $2A$ for $A=\begin{bmatrix}-1&3\\0&5\end{bmatrix}$.

   $\begin{bmatrix}-2&6\\0&10\end{bmatrix}$. Multiply each entry by $2$: $2(-1)=-2$, $2(3)=6$, $2(0)=0$, $2(5)=10$.

3. What are the dimensions of $AB$ if $A$ is $m \times n$ and $B$ is $n \times p$?

   $AB$ is $m \times p$. Product dimensions: outer dimensions of the factors.

4. What does the incidence matrix entry mean if column $e$ is edge $u\to v$ in a directed graph?

   In column $e$: $-1$ at row $u$, $+1$ at row $v$, and $0$ elsewhere. Shows edge direction: $-1$ at source, $+1$ at destination.

5. What is the inverse formula for a $2\times 2$ matrix $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ when invertible?

   $A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$. Formula uses determinant $ad-bc$ and swapped/negated entries.

6. Compute $AB$ for $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$ and $B=\begin{bmatrix}2\\-1\end{bmatrix}$.

   $\begin{bmatrix}0\\2\end{bmatrix}$. $1(2)+2(-1)=0$ and $3(2)+4(-1)=2$.

7. What do the entries $a_{ij}$ represent in a data matrix $A$ (rows and columns)?

   $a_{ij}$ is the value in row $i$ and column $j$ of $A$. Standard matrix notation: row index first, column index second.

8. What is the identity matrix $I_n$ and what key property does it satisfy with $A_{n\times n}$?

   $I_n$ has $1$ on the diagonal; $AI_n=I_nA=A$. Identity matrix acts as multiplicative identity for square matrices.

9. Identify the meaning of a payoff matrix entry $p_{ij}$ in a two-player game.

   $p_{ij}$ is the payoff when Player $1$ uses strategy $i$ and Player $2$ uses strategy $j$. Row player's payoff for their strategy choice against column player's choice.

10. Identify the matrix that leaves any compatible matrix $A$ unchanged under multiplication.

    The identity matrix $I$: $IA=A$ and $AI=A$. Identity matrix acts as multiplicative identity element.

11. Compute $AB$ for $A=\begin{bmatrix}1&2\end{bmatrix}$ and $B=\begin{bmatrix}3\\4\end{bmatrix}$.

    $AB=\begin{bmatrix}11\end{bmatrix}$. $1 \times 3 + 2 \times 4 = 3 + 8 = 11$.

12. What is the entry relationship between $A$ and its transpose $A^T$?

    $(A^T)_{ij}=a_{ji}$. Row $j$, column $i$ of $A$ becomes row $i$, column $j$ of $A^T$.

13. What does the row sum of an adjacency matrix represent in a directed network?

    Row sum of row $i$ equals the out-degree of node $i$. Counts edges leaving node $i$.

14. Compute $A+B$ for $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$ and $B=\begin{bmatrix}5&6\\7&8\end{bmatrix}$.

    $\begin{bmatrix}6&8\\10&12\end{bmatrix}$. Add corresponding entries: $1+5=6$, $2+6=8$, $3+7=10$, $4+8=12$.

15. What is the transpose of a matrix $A=[a_{ij}]$ in terms of entries?

    $A^T=[a_{ji}]$. Transpose swaps row and column indices.

16. What does the transpose $A^T$ do to the dimensions of $A_{m\times n}$?

    It swaps dimensions: $A^T$ is $n\times m$. Rows become columns and columns become rows.

17. What does the column sum of an adjacency matrix represent in a directed network?

    Column sum of column $j$ equals the in-degree of node $j$. Counts edges entering node $j$.

18. What is the condition on dimensions for the product $AB$ to be defined?

    If $A$ is $m \times n$ and $B$ is $n \times p$. Inner dimensions must match for matrix multiplication.

19. What does the entry $a_{ij}$ represent in a data matrix $A$?

    $a_{ij}$ is the value in row $i$, column $j$ of $A$. Standard notation for matrix entries uses row-column indexing.

20. Find $c_{12}$ for $A=\begin{bmatrix}2&1\\0&3\end{bmatrix}$ and $B=\begin{bmatrix}1&4\\2&-1\end{bmatrix}$ where $C=AB$.

    $c_{12}=7$. Row 1 of $A$ dot column 2 of $B$: $2(4)+1(-1)=7$.

21. Compute the row sums of $A=\begin{bmatrix}1&0&2\\3&-1&4\end{bmatrix}$ using $A\mathbf{1}$ with $\mathbf{1}=\begin{bmatrix}1\\1\\1\end{bmatrix}$.

    $A\mathbf{1}=\begin{bmatrix}3\\6\end{bmatrix}$. Multiplying by ones vector sums each row.

22. What is the entry formula for matrix addition: $(A+B)_{ij}$ equals what?

    $(A+B)_{ij}=a_{ij}+b_{ij}$. Add corresponding entries from the same position in each matrix.

23. What is the dot-product formula for an entry of a product matrix $C=AB$?

    $c_{ij}=\sum_{k=1}^{n} a_{ik}b_{kj}$. Row $i$ of $A$ dot column $j$ of $B$ gives entry $(i,j)$.

24. What scalar makes a $2\times 2$ matrix $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ non-invertible?

    Non-invertible when $ad-bc=0$. Determinant zero means matrix is singular.

25. For a payoff matrix $P$ and mixed strategies $r$ (rows) and $c$ (columns), what is the expected payoff formula?

    Expected payoff is $r^TPc$. Weighted average of payoffs using probability vectors.

26. What does multiplying a payoff matrix $P$ by a strategy vector $x$ compute in $Px$?

    $Px$ gives expected payoffs for each row strategy (given column mix $x$). Weighted average of payoffs based on opponent's strategy probabilities.

27. What is the entry formula for scalar multiplication: $(kA)_{ij}$ equals what?

    $(kA)_{ij}=k\,a_{ij}$. Multiply every entry by the scalar $k$.

28. Compute $A^T$ for $A=\begin{bmatrix}1&-2&0\\5&3&4\end{bmatrix}$.

    $\begin{bmatrix}1&5\\-2&3\\0&4\end{bmatrix}$. Swap rows and columns: first row becomes first column, etc.

29. What is the condition for a square matrix $A$ to be invertible using $A^{-1}$?

    There exists $A^{-1}$ with $AA^{-1}=I$ and $A^{-1}A=I$. Inverse undoes multiplication from either side.

30. What are the dimensions of a matrix with $m$ rows and $n$ columns?

    Dimensions are $m \times n$. Matrix size is always expressed as rows × columns.

31. What does the product $Ax$ represent when $A$ is a data matrix and $x$ is a weight vector?

    A weighted combination of columns of $A$ (a linear combination). Each column is weighted by corresponding entry in $x$.

32. What condition must be true about dimensions to add matrices $A_{m\times n}+B_{m\times n}$?

    They must have the same dimensions $m\times n$. Matrix addition requires element-wise operations, so dimensions must match.

33. What does the adjacency matrix entry $a_{ij}$ mean for a directed network?

    $a_{ij}=1$ means an edge from node $i$ to node $j$ (else $0$). Binary indicator: $1$ if directed edge exists, $0$ otherwise.

34. In a directed network incidence matrix, what do entries $-1$, $+1$, and $0$ mean for a node-edge pair?

    $-1$ leaves node, $+1$ enters node, $0$ not incident. Signs indicate edge direction relative to node.

35. What is the entry formula for a product matrix $C=AB$, specifically $c_{ij}$?

    $c_{ij}=\sum_{k=1}^{n} a_{ik}b_{kj}$. Dot product of row $i$ from $A$ with column $j$ from $B$.

36. In a payoff matrix, what does the entry $p_{ij}$ represent for row player strategy $i$ and column player strategy $j$?

    $p_{ij}$ is the payoff to the specified player for outcome $(i,j)$. Entry shows reward/cost for strategy combination.

37. What dimension condition is required to multiply $A_{m\times n}B_{n\times p}$?

    The inner dimensions must match: $A$ has $n$ columns and $B$ has $n$ rows. Number of columns in first matrix must equal rows in second.

38. What is the key property of transpose for a product $(AB)^T$?

    $(AB)^T=B^TA^T$. Order reverses when transposing a product.

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