Algebra 2 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 Algebra 2.

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: What is the coefficient matrix $A$ for $-x-2y=-3$ and $3x=12$?

Answer: $\begin{bmatrix}-1&-2\\3&0\end{bmatrix}$. First row $(-1,-2)$, second row $(3,0)$ from the equations.

Flashcard 2: What is the constant vector $\vec{b}$ for the system $2x-3y=5$ and $4x+y=-1$?

Answer: $\begin{bmatrix}5\\-1\end{bmatrix}$. Right-hand constants from both equations: $5$ and $-1$.

Flashcard 3: What is the defining feature of a homogeneous linear system in matrix form $A\vec{x}=\vec{b}$?

Answer: $\vec{b}=\vec{0}$. Zero vector on right side makes the system homogeneous.

Flashcard 4: What is the matrix equation for $7x-2y=0$ and $-5x+9y=0$ (a homogeneous system)?

Answer: $\begin{bmatrix}7&-2\\-5&9\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}$. Homogeneous system has all constants equal to zero.

Flashcard 5: What is $\vec{b}$ for $x+y+z=0$, $2x-y+3z=5$, $-x+4y-z=1$?

Answer: $\begin{bmatrix}0\\5\\1\end{bmatrix}$. Constants $0$, $5$, $1$ from right sides of the equations.

Flashcard 6: What is $A$ for $x+y+z=0$, $2x-y+3z=5$, $-x+4y-z=1$?

Answer: $\begin{bmatrix}1&1&1\\2&-1&3\\-1&4&-1\end{bmatrix}$. Coefficients from all three equations arranged by rows.

Flashcard 7: What is the matrix equation for $x+y+z=0$, $2x-y+3z=5$, and $-x+4y-z=1$?

Answer: $\begin{bmatrix}1&1&1\\2&-1&3\\-1&4&-1\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}0\\5\\1\end{bmatrix}$. Three equations in three variables forming $3\times 3$ system.

Flashcard 8: What is the constant vector $\vec{b}$ for $-x-2y=-3$ and $3x=12$?

Answer: $\begin{bmatrix}-3\\12\end{bmatrix}$. Right-hand constants $-3$ and $12$ from both equations.

Flashcard 9: What is the variable vector $\vec{x}$ for a system in variables $x$ and $y$ written as $A\vec{x}=\vec{b}$?

Answer: $\begin{bmatrix}x\\y\end{bmatrix}$. Variables $x$ and $y$ in column form for matrix multiplication.

Flashcard 10: What is the correct $A$ for $2x+y=5$ and $4x+3y=6$?

Answer: $\begin{bmatrix}2&1\\4&3\end{bmatrix}$. First row $(2,1)$, second row $(4,3)$ match equation coefficients.

Flashcard 11: Identify the error: using $\begin{bmatrix}2&1\\3&4\end{bmatrix}$ for $2x+y=5$ and $4x+3y=6$.

Answer: The second row should be $\begin{bmatrix}4&3\end{bmatrix}$, not $\begin{bmatrix}3&4\end{bmatrix}$. Second equation has coefficients $(4,3)$, not $(3,4)$.

Flashcard 12: Find the coefficient row for $y-3x=2$ when variable order is $x,y$.

Answer: $\begin{bmatrix}-3&1\end{bmatrix}$. Coefficients $(-3,1)$ when equation is written as $-3x+y=2$.

Flashcard 13: What is the size of $\vec{x}$ for a system of $m$ equations in $n$ variables written as $A\vec{x}=\vec{b}$?

Answer: $n\times 1$. $n$ variables stacked in one column.

Flashcard 14: What is the size of $\vec{b}$ for a system of $m$ equations written as $A\vec{x}=\vec{b}$?

Answer: $m\times 1$. $m$ constants stacked in one column.

Flashcard 15: What matrix equation represents the system $ax+by=c$ and $dx+ey=f$ using $A\vec{x}=\vec{b}$?

Answer: $\begin{bmatrix}a&b\\d&e\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}c\\f\end{bmatrix}$. First row has coefficients of first equation, second row has coefficients of second.

Flashcard 16: What is the coefficient matrix $A$ for the system $2x-3y=5$ and $4x+y=-1$?

Answer: $\begin{bmatrix}2&-3\\4&1\end{bmatrix}$. First row: $(2,-3)$, second row: $(4,1)$ from the two equations.

Flashcard 17: What matrix equation represents $y-3x=2$ and $5x+2y=1$ using variable order $x,y$?

Answer: $\begin{bmatrix}-3&1\\5&2\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}2\\1\end{bmatrix}$. Reorder $y-3x=2$ to $-3x+y=2$ for standard variable order.

Flashcard 18: What is the coefficient matrix $A$ for $3x-y+2z=4$ and $x+5y-z=6$?

Answer: $\begin{bmatrix}3&-1&2\\1&5&-1\end{bmatrix}$. Row 1: $(3,-1,2)$, row 2: $(1,5,-1)$ from equation coefficients.

Flashcard 19: What system of equations corresponds to $\begin{bmatrix}2&-1\\0&3\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}4\\9\end{bmatrix}$?

Answer: $2x-y=4$ and $3y=9$. First equation: $2x-y=4$; second equation: $3y=9$.

Flashcard 20: Identify $A$ and $\vec{b}$ for $\begin{bmatrix}1&2\\3&4\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}5\\6\end{bmatrix}$.

Answer: $A=\begin{bmatrix}1&2\\3&4\end{bmatrix},\ \vec{b}=\begin{bmatrix}5\\6\end{bmatrix}$. Coefficient matrix is the $2\times 2$ portion; constant vector is the right side.

Flashcard 21: What is the difference between $A\vec{x}=\vec{b}$ and the augmented matrix $[A\mid\vec{b}]$?

Answer: $A\vec{x}=\vec{b}$ is an equation; $[A\mid\vec{b}]$ is a single matrix. Matrix equation shows multiplication; augmented matrix shows combined data.

Flashcard 22: What is the augmented matrix corresponding to $A\vec{x}=\vec{b}$ if $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ and $\vec{b}=\begin{bmatrix}e\\f\end{bmatrix}$?

Answer: $\left[\begin{array}{cc|c}a&b&e\\c&d&f\end{array}\right]$. Augmented matrix combines $A$ and $\vec{b}$ with vertical separator.

Flashcard 23: What is the variable vector $\vec{x}$ for a system in variables $x$, $y$, and $z$ written as $A\vec{x}=\vec{b}$?

Answer: $\begin{bmatrix}x\\y\\z\end{bmatrix}$. All three variables $x$, $y$, $z$ stacked vertically.

Flashcard 24: What system corresponds to $\begin{bmatrix}1&0&-2\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}7\end{bmatrix}$?

Answer: $x-2z=7$. Single equation with three variables where $y$ coefficient is $0$.

Flashcard 25: What is the matrix equation for $2x+3y=1$ and $6x+9y=3$ written as $A\vec{x}=\vec{b}$?

Answer: $\begin{bmatrix}2&3\\6&9\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}1\\3\end{bmatrix}$. Standard form with proportional equations (second is $3$ times first).

Flashcard 26: Identify the matrix equation for $x+2y=7$ and $-3x+4y=1$ in the form $A\vec{x}=\vec{b}$.

Answer: $\begin{bmatrix}1&2\\-3&4\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}7\\1\end{bmatrix}$. Coefficients: row 1 is $(1,2)$, row 2 is $(-3,4)$; constants: $(7,1)$.

Flashcard 27: Identify the coefficient matrix $A$ for $4x=8$ and $-2y=10$ using variable order $x,y$.

Answer: $\begin{bmatrix}4&0\\0&-2\end{bmatrix}$. Diagonal matrix form for separate single-variable equations.

Flashcard 28: What is the matrix equation for the system $4x=8$ and $-2y=10$ using variable order $x,y$?

Answer: $\begin{bmatrix}4&0\\0&-2\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}8\\10\end{bmatrix}$. Diagonal matrix with coefficients $4$ and $-2$ on the diagonal.

Flashcard 29: What matrix is the coefficient matrix for $x=2$, $y=3$, $z=4$ written as $A\vec{x}=\vec{b}$?

Answer: $A=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$. Identity matrix has $1$s on diagonal, $0$s elsewhere.

Flashcard 30: What is the matrix equation form of a linear system with coefficient matrix $A$, variable vector $\vec{x}$, and constant vector $\vec{b}$?

Answer: $A\vec{x}=\vec{b}$. Standard form where $A$ multiplies variable vector $\vec{x}$ to equal constant vector $\vec{b}$.