Representing Linear Systems with Matrices - Precalculus

1. Identify the coefficient matrix $A$ for $x+0y-2z=7$ and $3x+y+z=4$.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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