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 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 2: What is the size of $A$ for a system of $3$ equations in $2$ variables written as $A\vec{x}=\vec{b}$?

Answer: $3\times 2$. Three equations (rows) by two variables (columns).

Flashcard 3: What does each row of $A$ correspond to when writing a system as $A\vec{x}=\vec{b}$?

Answer: One equation in the system. Each row of $A$ represents the coefficients from one equation.

Flashcard 4: 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 5: What is the matrix equation for the system $-2x+5y=9$ and $0x-3y=-6$?

Answer: $\begin{bmatrix}-2&5\\0&-3\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}9\\-6\end{bmatrix}$. First equation gives row $(-2,5)$, second gives row $(0,-3)$.

Flashcard 6: 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 7: What matrix equation represents the system $x-y=0$, $2x+3y=7$, $-x+4y=1$ as $A\vec{x}=\vec{b}$?

Answer: $\begin{bmatrix}1&-1\\2&3\\-1&4\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}0\\7\\1\end{bmatrix}$. Three equations in two variables creates $3\times 2$ coefficient matrix.

Flashcard 8: What does the coefficient matrix $A$ contain when writing $A\vec{x}=\vec{b}$ for a linear system?

Answer: All coefficients of the variables, arranged by equations. Each row contains coefficients from one equation in the system.

Flashcard 9: What does the constant vector $\vec{b}$ represent in $A\vec{x}=\vec{b}$?

Answer: The right-hand constants stacked in a column. Column vector containing constants from the right side of each equation.

Flashcard 10: 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 11: What is the coefficient matrix $A$ for $-2x+5y=9$ and $-3y=-6$?

Answer: $\begin{bmatrix}-2&5\\0&-3\end{bmatrix}$. Missing $x$ term in second equation gives coefficient $0$.

Flashcard 12: 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 13: 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 14: Identify the matrix equation for $\frac{1}{2}x+2y=3$ and $x-\frac{3}{4}y=1$.

Answer: $\begin{bmatrix}\frac{1}{2}&2\\1&-\frac{3}{4}\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}3\\1\end{bmatrix}$. Fractional coefficients are preserved in the matrix entries.

Flashcard 15: 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 16: 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 17: What does the variable vector $\vec{x}$ contain in a matrix equation $A\vec{x}=\vec{b}$?

Answer: The variables stacked in a column, such as $\begin{bmatrix}x\\y\end{bmatrix}$. Column vector with all variables from the system in order.

Flashcard 18: 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 $\vec{b}$.

Flashcard 19: 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 20: 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 21: What must you do first to write a system as $A\vec{x}=\vec{b}$ when equations are not aligned by variables?

Answer: Rewrite each equation in the same variable order. Ensures coefficient matrix has consistent variable ordering across rows.

Flashcard 22: Identify the matrix equation for $2x+z=3$ and $-x+4y=5$ using variable order $x,y,z$.

Answer: \begin{bmatrix}2&0&1\\-1&4&0\end{bmatrix} \begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}3\\5\end{bmatrix} $$. Missing $y$ in first equation and missing $z$ in second get coefficient $0$.

Flashcard 23: What is the coefficient row in $A$ for the equation $-x+4y=5$ using variable order $x,y,z$?

Answer: $\begin{bmatrix}-1&4&0\end{bmatrix}$. Coefficients $(-1,4,0)$ for variables $x$, $y$, $z$ respectively.

Flashcard 24: 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 25: What is the coefficient row in $A$ for the equation $2x+z=3$ using variable order $x,y,z$?

Answer: $\begin{bmatrix}2&0&1\end{bmatrix}$. Coefficients $(2,0,1)$ for variables $x$, $y$, $z$ respectively.

Flashcard 26: What is the constant vector $\vec{b}$ for $-2x+5y=9$ and $-3y=-6$?

Answer: $\begin{bmatrix}9\\-6\end{bmatrix}$. Constants $9$ and $-6$ from the right sides.

Flashcard 27: What does each column of $A$ correspond to when writing a system as $A\vec{x}=\vec{b}$?

Answer: One variable (in the chosen variable order). Each column represents coefficients of one variable across all equations.

Flashcard 28: 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 29: What is the matrix equation for $x+0y+0z=2$, $0x+y+0z=3$, $0x+0y+z=4$?

Answer: $\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}2\\3\\4\end{bmatrix}$. Identity matrix multiplied by variables equals the constant values.

Flashcard 30: What coefficient should be used in $A$ for a missing variable term, such as no $y$ term in an equation?

Answer: Use $0$ as the coefficient for that variable. Missing variables have coefficient $0$ in the matrix.

Flashcard 31: What is the constant vector $\vec{b}$ for $3x-y+2z=4$ and $x+5y-z=6$?

Answer: $\begin{bmatrix}4\\6\end{bmatrix}$. Constants $4$ and $6$ from right sides of both equations.

Flashcard 32: Which matrix equation matches the system $-x-2y=-3$ and $3x+0y=12$?

Answer: $\begin{bmatrix}-1&-2\\3&0\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}-3\\12\end{bmatrix}$. Negative coefficients and missing $y$ term in second equation.

Flashcard 33: Which vector is always a column vector in the matrix equation form $A\vec{x}=\vec{b}$?

Answer: $\vec{x}$ and $\vec{b}$ are column vectors. Both vectors are columns for proper matrix multiplication.

Flashcard 34: 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 35: 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 36: 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 37: 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 38: 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 39: What is the size of $A$ for a system of $2$ equations in $3$ variables written as $A\vec{x}=\vec{b}$?

Answer: $2\times 3$. Two equations (rows) by three variables (columns).

Flashcard 40: 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 41: 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 42: What is the size of $A$ for a system of $m$ equations in $n$ variables written as $A\vec{x}=\vec{b}$?

Answer: $m\times n$. $m$ rows (equations) by $n$ columns (variables).

Flashcard 43: 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 44: 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 45: 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 46: 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 47: 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 48: 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 49: What matrix equation represents $3x-y+2z=4$ and $x+5y-z=6$ as $A\vec{x}=\vec{b}$?

Answer: $\begin{bmatrix}3&-1&2\\1&5&-1\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}4\\6\end{bmatrix}$. Two equations with three variables each, forming $2\times 3$ coefficient matrix.

Flashcard 50: 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 51: 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 52: 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 53: 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.