Precalculus Flashcards: Representing Linear Systems With Matrices

What this deck covers

This deck focuses on Representing Linear Systems With Matrices, 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 the coefficient matrix $A$ for $x+0y-2z=7$ and $3x+y+z=4$.

Answer: $\begin{pmatrix}1&0&-2\\3&1&1\end{pmatrix}$. Missing $y$ term means coefficient is 0 in first row.

Flashcard 2: Find and correct $A$ for $2x+3y=1$ and $-x+4y=6$ if written as $A=\begin{bmatrix}2&3\\-1&-4\end{bmatrix}$.

Answer: $A=\begin{bmatrix}2&3\\-1&4\end{bmatrix}$. The coefficient 4 should be positive, not negative.

Flashcard 3: Identify the constants vector $\vec{b}$ for $2x-3y=5$ and $4x+y=-1$.

Answer: $\begin{pmatrix}5\\-1\end{pmatrix}$. Right-hand sides of the equations form the column vector.

Flashcard 4: Which number must be used as the coefficient of $y$ in $x-2z=7$ when forming matrix $A$?

Answer: $0$. Missing variables have coefficient 0 in the matrix.

Flashcard 5: What is the full matrix equation for $2x-3y=5$ and $4x+y=-1$ in the form $A\vec{x}=\vec{b}$?

Answer: $\begin{bmatrix}2&-3\\4&1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}5\\-1\end{bmatrix}$. Combines coefficient matrix, variable vector, and constants vector.

Flashcard 6: Identify the coefficient matrix $A$ for $2x-3y=5$ and $4x+y=-1$.

Answer: $\begin{pmatrix}2&-3\\4&1\end{pmatrix}$. Coefficients of $x$ and $y$ from each equation form the rows.

Flashcard 7: Which matrix is formed by appending $\vec{b}$ as a last column to $A$ for a linear system?

Answer: The augmented matrix $[A\mid \vec{b}]$. Combines $A$ and $\vec{b}$ with a vertical bar separator.

Flashcard 8: Identify the variable vector $\vec{x}$ for variables $x$ and $y$ in a matrix equation.

Answer: $\begin{pmatrix}x\\y\end{pmatrix}$. Variables arranged vertically in alphabetical order.

Flashcard 9: Identify the coefficient matrix $A$ for $2x-3y=5$ and $4x+y=-1$.

Answer: $\begin{bmatrix}2&-3\\4&1\end{bmatrix}$. Extract coefficients of $x$ and $y$ from each equation in order.

Flashcard 10: What is the matrix equation form of a linear system with coefficient matrix $A$, variable vector $\vec{x}$, and constants $\vec{b}$?

Answer: $A\vec{x}=\vec{b}$. Matrix multiplication of coefficient matrix and variable vector equals constants.

Flashcard 11: What does the variable vector $\vec{x}$ represent in $A\vec{x}=\vec{b}$?

Answer: A column vector of the system variables. Contains the unknowns $x_1, x_2, ..., x_n$ in a single column.

Flashcard 12: Which condition on $A$ guarantees a unique solution to $A\vec{x}=\vec{b}$ when $A$ is square?

Answer: $\det(A)\ne 0$. Non-zero determinant means the matrix is invertible.

Flashcard 13: What is the required order of variables in $\vec{x}$ when forming $A\vec{x}=\vec{b}$ from a system?

Answer: Use one consistent order in every equation (for example, $x,y,z$). Variable order must match across all equations and vectors.

Flashcard 14: What are the dimensions of $A$ for a system with $m$ equations and $n$ variables in $A\vec{x}=\vec{b}$?

Answer: $A$ is $m\times n$. Rows correspond to equations, columns to variables.

Flashcard 15: Write $A\vec{x}=\vec{b}$ for $2x-3y=5$ and $4x+y=-1$.

Answer: $\begin{pmatrix}2&-3\\4&1\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}5\\-1\end{pmatrix}$. Combines coefficient matrix, variables, and constants into one equation.

Flashcard 16: What are the dimensions of $\vec{x}$ for a system with $n$ variables?

Answer: $n\times 1$. Column vector with $n$ entries for $n$ variables.

Flashcard 17: What is the matrix equation form of a linear system with coefficient matrix $A$, variable vector $\vec{x}$, and constants $\vec{b}$?

Answer: $A\vec{x}=\vec{b}$. Matrix multiplication of coefficient matrix and variable vector equals constants.

Flashcard 18: What is the solution formula for $A\vec{x}=\vec{b}$ when $A$ is invertible?

Answer: $\vec{x}=A^{-1}\vec{b}$. Multiply both sides by $A^{-1}$ from the left to isolate $\vec{x}$.

Flashcard 19: What is the coefficient matrix $A$ for $a_1x+b_1y=c_1$ and $a_2x+b_2y=c_2$?

Answer: $A=\begin{bmatrix}a_1&b_1\\a_2&b_2\end{bmatrix}$. Coefficients are arranged row-wise, each row representing one equation.

Flashcard 20: Identify the variable vector $\vec{x}$ for variables $x$, $y$, and $z$ in order.

Answer: $\begin{pmatrix}x\\y\\z\end{pmatrix}$. Variables must be ordered consistently throughout.

Flashcard 21: What are the dimensions of $A$ for a system with $m$ equations and $n$ variables?

Answer: $m\times n$. $m$ rows for equations, $n$ columns for variables.

Flashcard 22: Identify $\vec{b}$ for the system $2x-3y=5$ and $4x+y=-1$.

Answer: $\begin{bmatrix}5\\-1\end{bmatrix}$. Right-hand side constants form the vector $\vec{b}$.

Flashcard 23: What do the entries of the coefficient matrix $A$ represent in $A\vec{x}=\vec{b}$?

Answer: The coefficients of the variables in the system. Each entry is the coefficient of a variable in an equation.

Flashcard 24: What does the constants vector $\vec{b}$ represent in $A\vec{x}=\vec{b}$?

Answer: A column vector of constant terms (right sides). Contains the values on the right side of each equation.

Flashcard 25: What are the dimensions of $\vec{x}$ and $\vec{b}$ for an $m\times n$ matrix $A$ in $A\vec{x}=\vec{b}$?

Answer: $\vec{x}$ is $n\times 1$ and $\vec{b}$ is $m\times 1$. Vector dimensions match system size for multiplication.

Flashcard 26: What is the coefficient matrix $A$ for $x+2y-3z=4$ and $-5x+y+z=0$?

Answer: $\begin{bmatrix}1&2&-3\\-5&1&1\end{bmatrix}$. Coefficients arranged for 2 equations with 3 variables.

Flashcard 27: Which matrix equation corresponds to the augmented matrix $\left[\begin{array}{cc|c}1&-2&3\\0&5&-10\end{array}\right]$?

Answer: $\begin{bmatrix}1&-2\\0&5\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}3\\-10\end{bmatrix}$. Read coefficients left of bar, constants right of bar.

Flashcard 28: Identify the augmented matrix $[A\mid\vec{b}]$ for $2x-3y=5$ and $4x+y=-1$.

Answer: $\left[\begin{array}{cc|c}2&-3&5\\4&1&-1\end{array}\right]$. Combines $A$ and $\vec{b}$ with a vertical bar separator.

Flashcard 29: What is the matrix equation for $x+2y-3z=4$ and $-5x+y+z=0$ written as $A\vec{x}=\vec{b}$?

Answer: $\begin{bmatrix}1&2&-3\\-5&1&1\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}4\\0\end{bmatrix}$. Matrix equation represents the given 2×3 system.

Flashcard 30: What is the vector variable for unknowns $x$ and $y$ when writing $A\vec{x}=\vec{b}$?

Answer: $\vec{x}=\begin{bmatrix}x\\y\end{bmatrix}$. Variables are arranged as a column vector in the same order as in equations.

Flashcard 31: What is the constants vector for equations with right-hand sides $c_1$ and $c_2$ in $A\vec{x}=\vec{b}$?

Answer: $\vec{b}=\begin{bmatrix}c_1\\c_2\end{bmatrix}$. Constants from the right side of equations form a column vector.

Flashcard 32: Identify $A$, $\vec{x}$, and $\vec{b}$ for $-7x=2$ and $3x=-5$ in $A\vec{x}=\vec{b}$.

Answer: $A=\begin{bmatrix}-7\\3\end{bmatrix},\ \vec{x}=\begin{bmatrix}x\end{bmatrix},\ \vec{b}=\begin{bmatrix}2\\-5\end{bmatrix}$. Single variable system has column vectors for coefficients.

Flashcard 33: State the relationship between $A\vec{x}=\vec{b}$ and the augmented matrix $[A\mid\vec{b}]$.

Answer: $[A\mid\vec{b}]$ is $A$ with $\vec{b}$ appended as last column. Augmented matrix combines all system information in one array.

Flashcard 34: What is the variable vector $\vec{x}$ for unknowns $x$, $y$, and $z$ in $A\vec{x}=\vec{b}$?

Answer: $\vec{x}=\begin{bmatrix}x\\y\\z\end{bmatrix}$. Three variables require a 3×1 column vector.

Flashcard 35: What are the dimensions of $\vec{b}$ for a system with $m$ equations?

Answer: $m\times 1$. Column vector with $m$ entries for $m$ equations.

Flashcard 36: What is the coefficient matrix $A$ for $3x+0y=7$ and $-2x+5y=1$?

Answer: $\begin{bmatrix}3&0\\-2&5\end{bmatrix}$. Zero coefficient indicates no $y$ term in the first equation.

Flashcard 37: Identify the augmented matrix $[A\mid \vec{b}]$ for $2x-3y=5$ and $4x+y=-1$.

Answer: $\left[\begin{array}{cc|c}2&-3&5\\4&1&-1\end{array}\right]$. Augmented matrix shows coefficients and constants together.

Flashcard 38: Which number is placed in $A$ when a variable is missing from an equation (for example, no $y$ term)?

Answer: Use coefficient $0$ for the missing variable. Missing variables are represented by zero coefficients.

Flashcard 39: Find and correct the matrix setup: $\begin{pmatrix}2&-3\\4&1\end{pmatrix}\begin{pmatrix}x&y\end{pmatrix}=\begin{pmatrix}5\\-1\end{pmatrix}$.

Answer: $\begin{pmatrix}2&-3\\4&1\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}5\\-1\end{pmatrix}$. $\vec{x}$ must be a column vector, not a row vector.