Using Matrices to Represent, Manipulate Data - Pre-Calculus
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What do the entries $a_{ij}$ represent in a data matrix $A$ (rows and columns)?
What do the entries $a_{ij}$ represent in a data matrix $A$ (rows and columns)?
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$a_{ij}$ is the value in row $i$ and column $j$ of $A$. Standard matrix notation: row index first, column index second.
$a_{ij}$ is the value in row $i$ and column $j$ of $A$. Standard matrix notation: row index first, column index second.
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What does multiplying a payoff matrix $P$ by a strategy vector $x$ compute in $Px$?
What does multiplying a payoff matrix $P$ by a strategy vector $x$ compute in $Px$?
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$Px$ gives expected payoffs for each row strategy (given column mix $x$). Weighted average of payoffs based on opponent's strategy probabilities.
$Px$ gives expected payoffs for each row strategy (given column mix $x$). Weighted average of payoffs based on opponent's strategy probabilities.
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Identify the meaning of a payoff matrix entry $p_{ij}$ in a two-player game.
Identify the meaning of a payoff matrix entry $p_{ij}$ in a two-player game.
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$p_{ij}$ 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.
$p_{ij}$ 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.
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What is the entry relationship between $A$ and its transpose $A^T$?
What is the entry relationship between $A$ and its transpose $A^T$?
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$(A^T){ij}=a{ji}$. Row $j$, column $i$ of $A$ becomes row $i$, column $j$ of $A^T$.
$(A^T){ij}=a{ji}$. Row $j$, column $i$ of $A$ becomes row $i$, column $j$ of $A^T$.
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What does the transpose $A^T$ do to the dimensions of $A_{m\times n}$?
What does the transpose $A^T$ do to the dimensions of $A_{m\times n}$?
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It swaps dimensions: $A^T$ is $n\times m$. Rows become columns and columns become rows.
It swaps dimensions: $A^T$ is $n\times m$. Rows become columns and columns become rows.
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What is the identity matrix $I_n$ and what key property does it satisfy with $A_{n\times n}$?
What is the identity matrix $I_n$ and what key property does it satisfy with $A_{n\times n}$?
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$I_n$ has $1$ on the diagonal; $AI_n=I_nA=A$. Identity matrix acts as multiplicative identity for square matrices.
$I_n$ has $1$ on the diagonal; $AI_n=I_nA=A$. Identity matrix acts as multiplicative identity for square matrices.
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What is the entry formula for a product matrix $C=AB$, specifically $c_{ij}$?
What is the entry formula for a product matrix $C=AB$, specifically $c_{ij}$?
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$c_{ij}=\sum_{k=1}^{n} a_{ik}b_{kj}$. Dot product of row $i$ from $A$ with column $j$ from $B$.
$c_{ij}=\sum_{k=1}^{n} a_{ik}b_{kj}$. Dot product of row $i$ from $A$ with column $j$ from $B$.
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What dimension condition is required to multiply $A_{m\times n}B_{n\times p}$?
What dimension condition is required to multiply $A_{m\times n}B_{n\times p}$?
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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.
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.
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What is the entry formula for scalar multiplication: $(kA)_{ij}$ equals what?
What is the entry formula for scalar multiplication: $(kA)_{ij}$ equals what?
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$(kA){ij}=k,a{ij}$. Multiply every entry by the scalar $k$.
$(kA){ij}=k,a{ij}$. Multiply every entry by the scalar $k$.
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What is the entry formula for matrix addition: $(A+B)_{ij}$ equals what?
What is the entry formula for matrix addition: $(A+B)_{ij}$ equals what?
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$(A+B){ij}=a{ij}+b_{ij}$. Add corresponding entries from the same position in each matrix.
$(A+B){ij}=a{ij}+b_{ij}$. Add corresponding entries from the same position in each matrix.
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What condition must be true about dimensions to add matrices $A_{m\times n}+B_{m\times n}$?
What condition must be true about dimensions to add matrices $A_{m\times n}+B_{m\times n}$?
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They must have the same dimensions $m\times n$. Matrix addition requires element-wise operations, so dimensions must match.
They must have the same dimensions $m\times n$. Matrix addition requires element-wise operations, so dimensions must match.
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Compute $A+B$ for $A=\begin{bmatrix}1&2\3&4\end{bmatrix}$ and $B=\begin{bmatrix}5&6\7&8\end{bmatrix}$.
Compute $A+B$ for $A=\begin{bmatrix}1&2\3&4\end{bmatrix}$ and $B=\begin{bmatrix}5&6\7&8\end{bmatrix}$.
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$\begin{bmatrix}6&8\10&12\end{bmatrix}$. Add corresponding entries: $1+5=6$, $2+6=8$, $3+7=10$, $4+8=12$.
$\begin{bmatrix}6&8\10&12\end{bmatrix}$. Add corresponding entries: $1+5=6$, $2+6=8$, $3+7=10$, $4+8=12$.
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Compute $A^T$ for $A=\begin{bmatrix}1&-2&0\5&3&4\end{bmatrix}$.
Compute $A^T$ for $A=\begin{bmatrix}1&-2&0\5&3&4\end{bmatrix}$.
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$\begin{bmatrix}1&5\-2&3\0&4\end{bmatrix}$. Swap rows and columns: first row becomes first column, etc.
$\begin{bmatrix}1&5\-2&3\0&4\end{bmatrix}$. Swap rows and columns: first row becomes first column, etc.
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Compute $AB$ for $A=\begin{bmatrix}1&2\3&4\end{bmatrix}$ and $B=\begin{bmatrix}2\-1\end{bmatrix}$.
Compute $AB$ for $A=\begin{bmatrix}1&2\3&4\end{bmatrix}$ and $B=\begin{bmatrix}2\-1\end{bmatrix}$.
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$\begin{bmatrix}0\2\end{bmatrix}$. $1(2)+2(-1)=0$ and $3(2)+4(-1)=2$.
$\begin{bmatrix}0\2\end{bmatrix}$. $1(2)+2(-1)=0$ and $3(2)+4(-1)=2$.
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Compute $2A$ for $A=\begin{bmatrix}-1&3\0&5\end{bmatrix}$.
Compute $2A$ for $A=\begin{bmatrix}-1&3\0&5\end{bmatrix}$.
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$\begin{bmatrix}-2&6\0&10\end{bmatrix}$. Multiply each entry by $2$: $2(-1)=-2$, $2(3)=6$, $2(0)=0$, $2(5)=10$.
$\begin{bmatrix}-2&6\0&10\end{bmatrix}$. Multiply each entry by $2$: $2(-1)=-2$, $2(3)=6$, $2(0)=0$, $2(5)=10$.
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What does the row sum of an adjacency matrix represent in a directed network?
What does the row sum of an adjacency matrix represent in a directed network?
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Row sum of row $i$ equals the out-degree of node $i$. Counts edges leaving node $i$.
Row sum of row $i$ equals the out-degree of node $i$. Counts edges leaving node $i$.
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What does the incidence matrix entry mean if column $e$ is edge $u\to v$ in a directed graph?
What does the incidence matrix entry mean if column $e$ is edge $u\to v$ in a directed graph?
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In column $e$: $-1$ at row $u$, $+1$ at row $v$, and $0$ elsewhere. Shows edge direction: $-1$ at source, $+1$ at destination.
In column $e$: $-1$ at row $u$, $+1$ at row $v$, and $0$ elsewhere. Shows edge direction: $-1$ at source, $+1$ at destination.
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What does the column sum of an adjacency matrix represent in a directed network?
What does the column sum of an adjacency matrix represent in a directed network?
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Column sum of column $j$ equals the in-degree of node $j$. Counts edges entering node $j$.
Column sum of column $j$ equals the in-degree of node $j$. Counts edges entering node $j$.
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What does the adjacency matrix entry $a_{ij}$ mean for a directed network?
What does the adjacency matrix entry $a_{ij}$ mean for a directed network?
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$a_{ij}=1$ means an edge from node $i$ to node $j$ (else $0$). Binary indicator: $1$ if directed edge exists, $0$ otherwise.
$a_{ij}=1$ means an edge from node $i$ to node $j$ (else $0$). Binary indicator: $1$ if directed edge exists, $0$ otherwise.
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What is the transpose of a matrix $A=[a_{ij}]$ in terms of entries?
What is the transpose of a matrix $A=[a_{ij}]$ in terms of entries?
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$A^T=[a_{ji}]$. Transpose swaps row and column indices.
$A^T=[a_{ji}]$. Transpose swaps row and column indices.
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Identify the matrix that leaves any compatible matrix $A$ unchanged under multiplication.
Identify the matrix that leaves any compatible matrix $A$ unchanged under multiplication.
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The identity matrix $I$: $IA=A$ and $AI=A$. Identity matrix acts as multiplicative identity element.
The identity matrix $I$: $IA=A$ and $AI=A$. Identity matrix acts as multiplicative identity element.
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What is the dot-product formula for an entry of a product matrix $C=AB$?
What is the dot-product formula for an entry of a product matrix $C=AB$?
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$c_{ij}=\sum_{k=1}^{n} a_{ik}b_{kj}$. Row $i$ of $A$ dot column $j$ of $B$ gives entry $(i,j)$.
$c_{ij}=\sum_{k=1}^{n} a_{ik}b_{kj}$. Row $i$ of $A$ dot column $j$ of $B$ gives entry $(i,j)$.
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What is the condition on dimensions for the product $AB$ to be defined?
What is the condition on dimensions for the product $AB$ to be defined?
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If $A$ is $m \times n$ and $B$ is $n \times p$. Inner dimensions must match for matrix multiplication.
If $A$ is $m \times n$ and $B$ is $n \times p$. Inner dimensions must match for matrix multiplication.
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What are the dimensions of a matrix with $m$ rows and $n$ columns?
What are the dimensions of a matrix with $m$ rows and $n$ columns?
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Dimensions are $m \times n$. Matrix size is always expressed as rows × columns.
Dimensions are $m \times n$. Matrix size is always expressed as rows × columns.
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What does the entry $a_{ij}$ represent in a data matrix $A$?
What does the entry $a_{ij}$ represent in a data matrix $A$?
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$a_{ij}$ is the value in row $i$, column $j$ of $A$. Standard notation for matrix entries uses row-column indexing.
$a_{ij}$ is the value in row $i$, column $j$ of $A$. Standard notation for matrix entries uses row-column indexing.
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What scalar makes a $2\times 2$ matrix $A=\begin{bmatrix}a&b\c&d\end{bmatrix}$ non-invertible?
What scalar makes a $2\times 2$ matrix $A=\begin{bmatrix}a&b\c&d\end{bmatrix}$ non-invertible?
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Non-invertible when $ad-bc=0$. Determinant zero means matrix is singular.
Non-invertible when $ad-bc=0$. Determinant zero means matrix is singular.
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What is the inverse formula for a $2\times 2$ matrix $A=\begin{bmatrix}a&b\c&d\end{bmatrix}$ when invertible?
What is the inverse formula for a $2\times 2$ matrix $A=\begin{bmatrix}a&b\c&d\end{bmatrix}$ when invertible?
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$A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\-c&a\end{bmatrix}$. Formula uses determinant $ad-bc$ and swapped/negated entries.
$A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\-c&a\end{bmatrix}$. Formula uses determinant $ad-bc$ and swapped/negated entries.
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What is the condition for a square matrix $A$ to be invertible using $A^{-1}$?
What is the condition for a square matrix $A$ to be invertible using $A^{-1}$?
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There exists $A^{-1}$ with $AA^{-1}=I$ and $A^{-1}A=I$. Inverse undoes multiplication from either side.
There exists $A^{-1}$ with $AA^{-1}=I$ and $A^{-1}A=I$. Inverse undoes multiplication from either side.
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What is the key property of transpose for a product $(AB)^T$?
What is the key property of transpose for a product $(AB)^T$?
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$(AB)^T=B^TA^T$. Order reverses when transposing a product.
$(AB)^T=B^TA^T$. Order reverses when transposing a product.
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Find $c_{12}$ for $A=\begin{bmatrix}2&1\0&3\end{bmatrix}$ and $B=\begin{bmatrix}1&4\2&-1\end{bmatrix}$ where $C=AB$.
Find $c_{12}$ for $A=\begin{bmatrix}2&1\0&3\end{bmatrix}$ and $B=\begin{bmatrix}1&4\2&-1\end{bmatrix}$ where $C=AB$.
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$c_{12}=7$. Row 1 of $A$ dot column 2 of $B$: $2(4)+1(-1)=7$.
$c_{12}=7$. Row 1 of $A$ dot column 2 of $B$: $2(4)+1(-1)=7$.
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