Precalculus Flashcards: Using Matrices To Represent Manipulate Data

What this deck covers

This deck focuses on Using Matrices To Represent Manipulate Data, giving you a quick way to review the definitions, rules, and examples that matter most for Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer: $p_{ij}$ 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Flashcard 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 $C=AB$.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer: 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.

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

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