Representing Linear Systems with Matrices - Pre-Calculus
Card 1 of 30
Which number must be used as the coefficient of $y$ in $x-2z=7$ when forming matrix $A$?
Which number must be used as the coefficient of $y$ in $x-2z=7$ when forming matrix $A$?
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$0$. Missing variables have coefficient 0 in the matrix.
$0$. Missing variables have coefficient 0 in the matrix.
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Identify the variable vector $\vec{x}$ for variables $x$, $y$, and $z$ in order.
Identify the variable vector $\vec{x}$ for variables $x$, $y$, and $z$ in order.
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$\begin{pmatrix}x\y\z\end{pmatrix}$. Variables must be ordered consistently throughout.
$\begin{pmatrix}x\y\z\end{pmatrix}$. Variables must be ordered consistently throughout.
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Identify the augmented matrix $[A\mid\vec{b}]$ for $2x-3y=5$ and $4x+y=-1$.
Identify the augmented matrix $[A\mid\vec{b}]$ for $2x-3y=5$ and $4x+y=-1$.
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$\left[\begin{array}{cc|c}2&-3&5\4&1&-1\end{array}\right]$. Combines $A$ and $\vec{b}$ with a vertical bar separator.
$\left[\begin{array}{cc|c}2&-3&5\4&1&-1\end{array}\right]$. Combines $A$ and $\vec{b}$ with a vertical bar separator.
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State the relationship between $A\vec{x}=\vec{b}$ and the augmented matrix $[A\mid\vec{b}]$.
State the relationship between $A\vec{x}=\vec{b}$ and the augmented matrix $[A\mid\vec{b}]$.
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$[A\mid\vec{b}]$ is $A$ with $\vec{b}$ appended as last column. Augmented matrix combines all system information in one array.
$[A\mid\vec{b}]$ is $A$ with $\vec{b}$ appended as last column. Augmented matrix combines all system information in one array.
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What is the matrix equation form of a linear system with coefficient matrix $A$, variable vector $\vec{x}$, and constants $\vec{b}$?
What is the matrix equation form of a linear system with coefficient matrix $A$, variable vector $\vec{x}$, and constants $\vec{b}$?
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$A\vec{x}=\vec{b}$. Matrix multiplication of coefficient matrix and variable vector equals constants.
$A\vec{x}=\vec{b}$. Matrix multiplication of coefficient matrix and variable vector equals constants.
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What do the entries of the coefficient matrix $A$ represent in $A\vec{x}=\vec{b}$?
What do the entries of the coefficient matrix $A$ represent in $A\vec{x}=\vec{b}$?
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The coefficients of the variables in the system. Each entry is the coefficient of a variable in an equation.
The coefficients of the variables in the system. Each entry is the coefficient of a variable in an equation.
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What does the variable vector $\vec{x}$ represent in $A\vec{x}=\vec{b}$?
What does the variable vector $\vec{x}$ represent in $A\vec{x}=\vec{b}$?
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A column vector of the system variables. Contains the unknowns $x_1, x_2, ..., x_n$ in a single column.
A column vector of the system variables. Contains the unknowns $x_1, x_2, ..., x_n$ in a single column.
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What does the constants vector $\vec{b}$ represent in $A\vec{x}=\vec{b}$?
What does the constants vector $\vec{b}$ represent in $A\vec{x}=\vec{b}$?
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A column vector of constant terms (right sides). Contains the values on the right side of each equation.
A column vector of constant terms (right sides). Contains the values on the right side of each equation.
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What are the dimensions of $A$ for a system with $m$ equations and $n$ variables?
What are the dimensions of $A$ for a system with $m$ equations and $n$ variables?
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$m\times n$. $m$ rows for equations, $n$ columns for variables.
$m\times n$. $m$ rows for equations, $n$ columns for variables.
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What are the dimensions of $\vec{x}$ for a system with $n$ variables?
What are the dimensions of $\vec{x}$ for a system with $n$ variables?
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$n\times 1$. Column vector with $n$ entries for $n$ variables.
$n\times 1$. Column vector with $n$ entries for $n$ variables.
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What are the dimensions of $\vec{b}$ for a system with $m$ equations?
What are the dimensions of $\vec{b}$ for a system with $m$ equations?
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$m\times 1$. Column vector with $m$ entries for $m$ equations.
$m\times 1$. Column vector with $m$ entries for $m$ equations.
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Identify the coefficient matrix $A$ for $2x-3y=5$ and $4x+y=-1$.
Identify the coefficient matrix $A$ for $2x-3y=5$ and $4x+y=-1$.
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$\begin{pmatrix}2&-3\4&1\end{pmatrix}$. Coefficients of $x$ and $y$ from each equation form the rows.
$\begin{pmatrix}2&-3\4&1\end{pmatrix}$. Coefficients of $x$ and $y$ from each equation form the rows.
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Identify the variable vector $\vec{x}$ for variables $x$ and $y$ in a matrix equation.
Identify the variable vector $\vec{x}$ for variables $x$ and $y$ in a matrix equation.
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$\begin{pmatrix}x\y\end{pmatrix}$. Variables arranged vertically in alphabetical order.
$\begin{pmatrix}x\y\end{pmatrix}$. Variables arranged vertically in alphabetical order.
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Identify the constants vector $\vec{b}$ for $2x-3y=5$ and $4x+y=-1$.
Identify the constants vector $\vec{b}$ for $2x-3y=5$ and $4x+y=-1$.
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$\begin{pmatrix}5\-1\end{pmatrix}$. Right-hand sides of the equations form the column vector.
$\begin{pmatrix}5\-1\end{pmatrix}$. Right-hand sides of the equations form the column vector.
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Write $A\vec{x}=\vec{b}$ for $2x-3y=5$ and $4x+y=-1$.
Write $A\vec{x}=\vec{b}$ for $2x-3y=5$ and $4x+y=-1$.
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$\begin{pmatrix}2&-3\4&1\end{pmatrix}\begin{pmatrix}x\y\end{pmatrix}=\begin{pmatrix}5\-1\end{pmatrix}$. Combines coefficient matrix, variables, and constants into one equation.
$\begin{pmatrix}2&-3\4&1\end{pmatrix}\begin{pmatrix}x\y\end{pmatrix}=\begin{pmatrix}5\-1\end{pmatrix}$. Combines coefficient matrix, variables, and constants into one equation.
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Identify the coefficient matrix $A$ for $x+0y-2z=7$ and $3x+y+z=4$.
Identify the coefficient matrix $A$ for $x+0y-2z=7$ and $3x+y+z=4$.
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$\begin{pmatrix}1&0&-2\3&1&1\end{pmatrix}$. Missing $y$ term means coefficient is 0 in first row.
$\begin{pmatrix}1&0&-2\3&1&1\end{pmatrix}$. Missing $y$ term means coefficient is 0 in first row.
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Find and correct the matrix setup: $\begin{pmatrix}2&-3\4&1\end{pmatrix}\begin{pmatrix}x&y\end{pmatrix}=\begin{pmatrix}5\-1\end{pmatrix}$.
Find and correct the matrix setup: $\begin{pmatrix}2&-3\4&1\end{pmatrix}\begin{pmatrix}x&y\end{pmatrix}=\begin{pmatrix}5\-1\end{pmatrix}$.
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$\begin{pmatrix}2&-3\4&1\end{pmatrix}\begin{pmatrix}x\y\end{pmatrix}=\begin{pmatrix}5\-1\end{pmatrix}$. $\vec{x}$ must be a column vector, not a row vector.
$\begin{pmatrix}2&-3\4&1\end{pmatrix}\begin{pmatrix}x\y\end{pmatrix}=\begin{pmatrix}5\-1\end{pmatrix}$. $\vec{x}$ must be a column vector, not a row vector.
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Which condition on $A$ guarantees a unique solution to $A\vec{x}=\vec{b}$ when $A$ is square?
Which condition on $A$ guarantees a unique solution to $A\vec{x}=\vec{b}$ when $A$ is square?
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$\det(A)\ne 0$. Non-zero determinant means the matrix is invertible.
$\det(A)\ne 0$. Non-zero determinant means the matrix is invertible.
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What is the solution formula for $A\vec{x}=\vec{b}$ when $A$ is invertible?
What is the solution formula for $A\vec{x}=\vec{b}$ when $A$ is invertible?
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$\vec{x}=A^{-1}\vec{b}$. Multiply both sides by $A^{-1}$ from the left to isolate $\vec{x}$.
$\vec{x}=A^{-1}\vec{b}$. Multiply both sides by $A^{-1}$ from the left to isolate $\vec{x}$.
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What is the matrix equation form of a linear system with coefficient matrix $A$, variable vector $\vec{x}$, and constants $\vec{b}$?
What is the matrix equation form of a linear system with coefficient matrix $A$, variable vector $\vec{x}$, and constants $\vec{b}$?
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$A\vec{x}=\vec{b}$. Matrix multiplication of coefficient matrix and variable vector equals constants.
$A\vec{x}=\vec{b}$. Matrix multiplication of coefficient matrix and variable vector equals constants.
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What is the vector variable for unknowns $x$ and $y$ when writing $A\vec{x}=\vec{b}$?
What is the vector variable for unknowns $x$ and $y$ when writing $A\vec{x}=\vec{b}$?
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$\vec{x}=\begin{bmatrix}x\y\end{bmatrix}$. Variables are arranged as a column vector in the same order as in equations.
$\vec{x}=\begin{bmatrix}x\y\end{bmatrix}$. Variables are arranged as a column vector in the same order as in equations.
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What is the constants vector for equations with right-hand sides $c_1$ and $c_2$ in $A\vec{x}=\vec{b}$?
What is the constants vector for equations with right-hand sides $c_1$ and $c_2$ in $A\vec{x}=\vec{b}$?
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$\vec{b}=\begin{bmatrix}c_1\c_2\end{bmatrix}$. Constants from the right side of equations form a column vector.
$\vec{b}=\begin{bmatrix}c_1\c_2\end{bmatrix}$. Constants from the right side of equations form a column vector.
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Identify $\vec{b}$ for the system $2x-3y=5$ and $4x+y=-1$.
Identify $\vec{b}$ for the system $2x-3y=5$ and $4x+y=-1$.
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$\begin{bmatrix}5\-1\end{bmatrix}$. Right-hand side constants form the vector $\vec{b}$.
$\begin{bmatrix}5\-1\end{bmatrix}$. Right-hand side constants form the vector $\vec{b}$.
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What is the full matrix equation for $2x-3y=5$ and $4x+y=-1$ in the form $A\vec{x}=\vec{b}$?
What is the full matrix equation for $2x-3y=5$ and $4x+y=-1$ in the form $A\vec{x}=\vec{b}$?
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$\begin{bmatrix}2&-3\4&1\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}5\-1\end{bmatrix}$. Combines coefficient matrix, variable vector, and constants vector.
$\begin{bmatrix}2&-3\4&1\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}5\-1\end{bmatrix}$. Combines coefficient matrix, variable vector, and constants vector.
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What is the coefficient matrix $A$ for $3x+0y=7$ and $-2x+5y=1$?
What is the coefficient matrix $A$ for $3x+0y=7$ and $-2x+5y=1$?
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$\begin{bmatrix}3&0\-2&5\end{bmatrix}$. Zero coefficient indicates no $y$ term in the first equation.
$\begin{bmatrix}3&0\-2&5\end{bmatrix}$. Zero coefficient indicates no $y$ term in the first equation.
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What is the coefficient matrix $A$ for $x+2y-3z=4$ and $-5x+y+z=0$?
What is the coefficient matrix $A$ for $x+2y-3z=4$ and $-5x+y+z=0$?
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$\begin{bmatrix}1&2&-3\-5&1&1\end{bmatrix}$. Coefficients arranged for 2 equations with 3 variables.
$\begin{bmatrix}1&2&-3\-5&1&1\end{bmatrix}$. Coefficients arranged for 2 equations with 3 variables.
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What is the variable vector $\vec{x}$ for unknowns $x$, $y$, and $z$ in $A\vec{x}=\vec{b}$?
What is the variable vector $\vec{x}$ for unknowns $x$, $y$, and $z$ in $A\vec{x}=\vec{b}$?
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$\vec{x}=\begin{bmatrix}x\y\z\end{bmatrix}$. Three variables require a 3×1 column vector.
$\vec{x}=\begin{bmatrix}x\y\z\end{bmatrix}$. Three variables require a 3×1 column vector.
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What is the matrix equation for $x+2y-3z=4$ and $-5x+y+z=0$ written as $A\vec{x}=\vec{b}$?
What is the matrix equation for $x+2y-3z=4$ and $-5x+y+z=0$ written as $A\vec{x}=\vec{b}$?
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$\begin{bmatrix}1&2&-3\-5&1&1\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}4\0\end{bmatrix}$. Matrix equation represents the given 2×3 system.
$\begin{bmatrix}1&2&-3\-5&1&1\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}4\0\end{bmatrix}$. Matrix equation represents the given 2×3 system.
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What is the coefficient matrix $A$ for $a_1x+b_1y=c_1$ and $a_2x+b_2y=c_2$?
What is the coefficient matrix $A$ for $a_1x+b_1y=c_1$ and $a_2x+b_2y=c_2$?
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$A=\begin{bmatrix}a_1&b_1\a_2&b_2\end{bmatrix}$. Coefficients are arranged row-wise, each row representing one equation.
$A=\begin{bmatrix}a_1&b_1\a_2&b_2\end{bmatrix}$. Coefficients are arranged row-wise, each row representing one equation.
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What is the required order of variables in $\vec{x}$ when forming $A\vec{x}=\vec{b}$ from a system?
What is the required order of variables in $\vec{x}$ when forming $A\vec{x}=\vec{b}$ from a system?
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Use one consistent order in every equation (for example, $x,y,z$). Variable order must match across all equations and vectors.
Use one consistent order in every equation (for example, $x,y,z$). Variable order must match across all equations and vectors.
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