Representing Linear Systems with Matrices - Algebra 2
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What is the coefficient matrix $A$ for $-x-2y=-3$ and $3x=12$?
What is the coefficient matrix $A$ for $-x-2y=-3$ and $3x=12$?
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$\begin{bmatrix}-1&-2\3&0\end{bmatrix}$. First row $(-1,-2)$, second row $(3,0)$ from the equations.
$\begin{bmatrix}-1&-2\3&0\end{bmatrix}$. First row $(-1,-2)$, second row $(3,0)$ from the equations.
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What is the constant vector $\vec{b}$ for the system $2x-3y=5$ and $4x+y=-1$?
What is the constant vector $\vec{b}$ for the system $2x-3y=5$ and $4x+y=-1$?
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$\begin{bmatrix}5\-1\end{bmatrix}$. Right-hand constants from both equations: $5$ and $-1$.
$\begin{bmatrix}5\-1\end{bmatrix}$. Right-hand constants from both equations: $5$ and $-1$.
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What is the defining feature of a homogeneous linear system in matrix form $A\vec{x}=\vec{b}$?
What is the defining feature of a homogeneous linear system in matrix form $A\vec{x}=\vec{b}$?
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$\vec{b}=\vec{0}$. Zero vector on right side makes the system homogeneous.
$\vec{b}=\vec{0}$. Zero vector on right side makes the system homogeneous.
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What is the matrix equation for $7x-2y=0$ and $-5x+9y=0$ (a homogeneous system)?
What is the matrix equation for $7x-2y=0$ and $-5x+9y=0$ (a homogeneous system)?
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$\begin{bmatrix}7&-2\-5&9\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}0\0\end{bmatrix}$. Homogeneous system has all constants equal to zero.
$\begin{bmatrix}7&-2\-5&9\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}0\0\end{bmatrix}$. Homogeneous system has all constants equal to zero.
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What is $\vec{b}$ for $x+y+z=0$, $2x-y+3z=5$, $-x+4y-z=1$?
What is $\vec{b}$ for $x+y+z=0$, $2x-y+3z=5$, $-x+4y-z=1$?
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$\begin{bmatrix}0\5\1\end{bmatrix}$. Constants $0$, $5$, $1$ from right sides of the equations.
$\begin{bmatrix}0\5\1\end{bmatrix}$. Constants $0$, $5$, $1$ from right sides of the equations.
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What is $A$ for $x+y+z=0$, $2x-y+3z=5$, $-x+4y-z=1$?
What is $A$ for $x+y+z=0$, $2x-y+3z=5$, $-x+4y-z=1$?
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$\begin{bmatrix}1&1&1\2&-1&3\-1&4&-1\end{bmatrix}$. Coefficients from all three equations arranged by rows.
$\begin{bmatrix}1&1&1\2&-1&3\-1&4&-1\end{bmatrix}$. Coefficients from all three equations arranged by rows.
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What is the matrix equation for $x+y+z=0$, $2x-y+3z=5$, and $-x+4y-z=1$?
What is the matrix equation for $x+y+z=0$, $2x-y+3z=5$, and $-x+4y-z=1$?
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$\begin{bmatrix}1&1&1\2&-1&3\-1&4&-1\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}0\5\1\end{bmatrix}$. Three equations in three variables forming $3\times 3$ system.
$\begin{bmatrix}1&1&1\2&-1&3\-1&4&-1\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}0\5\1\end{bmatrix}$. Three equations in three variables forming $3\times 3$ system.
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What is the constant vector $\vec{b}$ for $-x-2y=-3$ and $3x=12$?
What is the constant vector $\vec{b}$ for $-x-2y=-3$ and $3x=12$?
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$\begin{bmatrix}-3\12\end{bmatrix}$. Right-hand constants $-3$ and $12$ from both equations.
$\begin{bmatrix}-3\12\end{bmatrix}$. Right-hand constants $-3$ and $12$ from both equations.
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What is the variable vector $\vec{x}$ for a system in variables $x$ and $y$ written as $A\vec{x}=\vec{b}$?
What is the variable vector $\vec{x}$ for a system in variables $x$ and $y$ written as $A\vec{x}=\vec{b}$?
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$\begin{bmatrix}x\y\end{bmatrix}$. Variables $x$ and $y$ in column form for matrix multiplication.
$\begin{bmatrix}x\y\end{bmatrix}$. Variables $x$ and $y$ in column form for matrix multiplication.
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What is the correct $A$ for $2x+y=5$ and $4x+3y=6$?
What is the correct $A$ for $2x+y=5$ and $4x+3y=6$?
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$\begin{bmatrix}2&1\4&3\end{bmatrix}$. First row $(2,1)$, second row $(4,3)$ match equation coefficients.
$\begin{bmatrix}2&1\4&3\end{bmatrix}$. First row $(2,1)$, second row $(4,3)$ match equation coefficients.
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Identify the error: using $\begin{bmatrix}2&1\3&4\end{bmatrix}$ for $2x+y=5$ and $4x+3y=6$.
Identify the error: using $\begin{bmatrix}2&1\3&4\end{bmatrix}$ for $2x+y=5$ and $4x+3y=6$.
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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 $(3,4)$.
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 $(3,4)$.
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Find the coefficient row for $y-3x=2$ when variable order is $x,y$.
Find the coefficient row for $y-3x=2$ when variable order is $x,y$.
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$\begin{bmatrix}-3&1\end{bmatrix}$. Coefficients $(-3,1)$ when equation is written as $-3x+y=2$.
$\begin{bmatrix}-3&1\end{bmatrix}$. Coefficients $(-3,1)$ when equation is written as $-3x+y=2$.
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What is the size of $\vec{x}$ for a system of $m$ equations in $n$ variables written as $A\vec{x}=\vec{b}$?
What is the size of $\vec{x}$ for a system of $m$ equations in $n$ variables written as $A\vec{x}=\vec{b}$?
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$n\times 1$. $n$ variables stacked in one column.
$n\times 1$. $n$ variables stacked in one column.
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What is the size of $\vec{b}$ for a system of $m$ equations written as $A\vec{x}=\vec{b}$?
What is the size of $\vec{b}$ for a system of $m$ equations written as $A\vec{x}=\vec{b}$?
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$m\times 1$. $m$ constants stacked in one column.
$m\times 1$. $m$ constants stacked in one column.
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What matrix equation represents the system $ax+by=c$ and $dx+ey=f$ using $A\vec{x}=\vec{b}$?
What matrix equation represents the system $ax+by=c$ and $dx+ey=f$ using $A\vec{x}=\vec{b}$?
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$\begin{bmatrix}a&b\d&e\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}c\f\end{bmatrix}$. First row has coefficients of first equation, second row has coefficients of second.
$\begin{bmatrix}a&b\d&e\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}c\f\end{bmatrix}$. First row has coefficients of first equation, second row has coefficients of second.
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What is the coefficient matrix $A$ for the system $2x-3y=5$ and $4x+y=-1$?
What is the coefficient matrix $A$ for the system $2x-3y=5$ and $4x+y=-1$?
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$\begin{bmatrix}2&-3\4&1\end{bmatrix}$. First row: $(2,-3)$, second row: $(4,1)$ from the two equations.
$\begin{bmatrix}2&-3\4&1\end{bmatrix}$. First row: $(2,-3)$, second row: $(4,1)$ from the two equations.
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What matrix equation represents $y-3x=2$ and $5x+2y=1$ using variable order $x,y$?
What matrix equation represents $y-3x=2$ and $5x+2y=1$ using variable order $x,y$?
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$$\begin{bmatrix}-3&1\5&2\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}2\1\end{bmatrix}$$. Reorder $y-3x=2$ to $-3x+y=2$ for standard variable order.
$$\begin{bmatrix}-3&1\5&2\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}2\1\end{bmatrix}$$. Reorder $y-3x=2$ to $-3x+y=2$ for standard variable order.
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What is the coefficient matrix $A$ for $3x-y+2z=4$ and $x+5y-z=6$?
What is the coefficient matrix $A$ for $3x-y+2z=4$ and $x+5y-z=6$?
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$\begin{bmatrix}3&-1&2\1&5&-1\end{bmatrix}$. Row 1: $(3,-1,2)$, row 2: $(1,5,-1)$ from equation coefficients.
$\begin{bmatrix}3&-1&2\1&5&-1\end{bmatrix}$. Row 1: $(3,-1,2)$, row 2: $(1,5,-1)$ from equation coefficients.
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What system of equations corresponds to $\begin{bmatrix}2&-1\0&3\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}4\9\end{bmatrix}$?
What system of equations corresponds to $\begin{bmatrix}2&-1\0&3\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}4\9\end{bmatrix}$?
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$2x-y=4$ and $3y=9$. First equation: $2x-y=4$; second equation: $3y=9$.
$2x-y=4$ and $3y=9$. First equation: $2x-y=4$; second equation: $3y=9$.
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Identify $A$ and $\vec{b}$ for $\begin{bmatrix}1&2\3&4\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}5\6\end{bmatrix}$.
Identify $A$ and $\vec{b}$ for $\begin{bmatrix}1&2\3&4\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}5\6\end{bmatrix}$.
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$A=\begin{bmatrix}1&2\3&4\end{bmatrix},\\ \vec{b}=\begin{bmatrix}5\6\end{bmatrix}$. Coefficient matrix is the $2\times 2$ portion; constant vector is the right side.
$A=\begin{bmatrix}1&2\3&4\end{bmatrix},\\ \vec{b}=\begin{bmatrix}5\6\end{bmatrix}$. Coefficient matrix is the $2\times 2$ portion; constant vector is the right side.
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What is the difference between $A\vec{x}=\vec{b}$ and the augmented matrix $[A\mid\vec{b}]$?
What is the difference between $A\vec{x}=\vec{b}$ and the augmented matrix $[A\mid\vec{b}]$?
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$A\vec{x}=\vec{b}$ is an equation; $[A\mid\vec{b}]$ is a single matrix. Matrix equation shows multiplication; augmented matrix shows combined data.
$A\vec{x}=\vec{b}$ is an equation; $[A\mid\vec{b}]$ is a single matrix. Matrix equation shows multiplication; augmented matrix shows combined data.
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What is the augmented matrix corresponding to $A\vec{x}=\vec{b}$ if $A=\begin{bmatrix}a&b\c&d\end{bmatrix}$ and $\vec{b}=\begin{bmatrix}e\f\end{bmatrix}$?
What is the augmented matrix corresponding to $A\vec{x}=\vec{b}$ if $A=\begin{bmatrix}a&b\c&d\end{bmatrix}$ and $\vec{b}=\begin{bmatrix}e\f\end{bmatrix}$?
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$\left[\begin{array}{cc|c}a&b&e\c&d&f\end{array}\right]$. Augmented matrix combines $A$ and $\vec{b}$ with vertical separator.
$\left[\begin{array}{cc|c}a&b&e\c&d&f\end{array}\right]$. Augmented matrix combines $A$ and $\vec{b}$ with vertical separator.
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What is the variable vector $\vec{x}$ for a system in variables $x$, $y$, and $z$ written as $A\vec{x}=\vec{b}$?
What is the variable vector $\vec{x}$ for a system in variables $x$, $y$, and $z$ written as $A\vec{x}=\vec{b}$?
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$\begin{bmatrix}x\y\z\end{bmatrix}$. All three variables $x$, $y$, $z$ stacked vertically.
$\begin{bmatrix}x\y\z\end{bmatrix}$. All three variables $x$, $y$, $z$ stacked vertically.
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What system corresponds to $\begin{bmatrix}1&0&-2\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}7\end{bmatrix}$?
What system corresponds to $\begin{bmatrix}1&0&-2\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}7\end{bmatrix}$?
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$x-2z=7$. Single equation with three variables where $y$ coefficient is $0$.
$x-2z=7$. Single equation with three variables where $y$ coefficient is $0$.
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What is the matrix equation for $2x+3y=1$ and $6x+9y=3$ written as $A\vec{x}=\vec{b}$?
What is the matrix equation for $2x+3y=1$ and $6x+9y=3$ written as $A\vec{x}=\vec{b}$?
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$\begin{bmatrix}2&3\6&9\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}1\3\end{bmatrix}$. Standard form with proportional equations (second is $3$ times first).
$\begin{bmatrix}2&3\6&9\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}1\3\end{bmatrix}$. Standard form with proportional equations (second is $3$ times first).
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Identify the matrix equation for $x+2y=7$ and $-3x+4y=1$ in the form $A\vec{x}=\vec{b}$.
Identify the matrix equation for $x+2y=7$ and $-3x+4y=1$ in the form $A\vec{x}=\vec{b}$.
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$\begin{bmatrix}1&2\-3&4\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}7\1\end{bmatrix}$. Coefficients: row 1 is $(1,2)$, row 2 is $(-3,4)$; constants: $(7,1)$.
$\begin{bmatrix}1&2\-3&4\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}7\1\end{bmatrix}$. Coefficients: row 1 is $(1,2)$, row 2 is $(-3,4)$; constants: $(7,1)$.
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Identify the coefficient matrix $A$ for $4x=8$ and $-2y=10$ using variable order $x,y$.
Identify the coefficient matrix $A$ for $4x=8$ and $-2y=10$ using variable order $x,y$.
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$\begin{bmatrix}4&0\0&-2\end{bmatrix}$. Diagonal matrix form for separate single-variable equations.
$\begin{bmatrix}4&0\0&-2\end{bmatrix}$. Diagonal matrix form for separate single-variable equations.
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What is the matrix equation for the system $4x=8$ and $-2y=10$ using variable order $x,y$?
What is the matrix equation for the system $4x=8$ and $-2y=10$ using variable order $x,y$?
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$\begin{bmatrix}4&0\0&-2\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}8\10\end{bmatrix}$. Diagonal matrix with coefficients $4$ and $-2$ on the diagonal.
$\begin{bmatrix}4&0\0&-2\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}8\10\end{bmatrix}$. Diagonal matrix with coefficients $4$ and $-2$ on the diagonal.
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What matrix is the coefficient matrix for $x=2$, $y=3$, $z=4$ written as $A\vec{x}=\vec{b}$?
What matrix is the coefficient matrix for $x=2$, $y=3$, $z=4$ written as $A\vec{x}=\vec{b}$?
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$A=\begin{bmatrix}1&0&0\0&1&0\0&0&1\end{bmatrix}$. Identity matrix has $1$s on diagonal, $0$s elsewhere.
$A=\begin{bmatrix}1&0&0\0&1&0\0&0&1\end{bmatrix}$. Identity matrix has $1$s on diagonal, $0$s elsewhere.
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What is the matrix equation form of a linear system with coefficient matrix $A$, variable vector $\vec{x}$, and constant vector $\vec{b}$?
What is the matrix equation form of a linear system with coefficient matrix $A$, variable vector $\vec{x}$, and constant vector $\vec{b}$?
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$A\vec{x}=\vec{b}$. Standard form where $A$ multiplies variable vector $\vec{x}$ to equal constant vector $\vec{b}$.
$A\vec{x}=\vec{b}$. Standard form where $A$ multiplies variable vector $\vec{x}$ to equal constant vector $\vec{b}$.
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