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Linear Algebra : Row Operations and Equivalent Systems

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Example Questions

Example Question #71 : Linear Equations

$\begin{align*}&\text{Identify the matrix which is not a equivalent to:}\\&\begin{bmatrix}19&6&-10\\-1&-12&7\\\end{bmatrix}\end{align*}$

Possible Answers:

$\begin{bmatrix}\frac{19}{2}&3&-5\\\frac{1}{6}&2&-\frac{7}{6}\\\end{bmatrix}$

$\begin{bmatrix}-38&-12&20\\2&24&-14\\\end{bmatrix}$

$\begin{bmatrix}20&18&-17\\-1&-12&7\\\end{bmatrix}$

$\begin{bmatrix}22&9&-7\\2&-9&10\\\end{bmatrix}$

Correct answer:

$\begin{bmatrix}22&9&-7\\2&-9&10\\\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Two matrices are equivalent if one can be turned into another}\\&\text{via row operations: scaling a row by multiplication of some}\\&\text{constant, addition/subtraction of one row with another, or a}\\&\text{combination of these processes.}\\&\text{The matrix: }\begin{bmatrix}-38&-12&20\\2&24&-14\\\end{bmatrix}\text{ is our entire initial matrix scaled by }-2.\\&\text{The matrix }\begin{bmatrix}\frac{19}{2}&3&-5\\\frac{1}{6}&2&-\frac{7}{6}\\\end{bmatrix}\text{ can be formed by scaling the first and second rows of}\\&\text{the orignal by }\frac{1}{2}\text{ and }-\frac{1}{6}.\\&\text{Finally, subtracting one row from another creates }\begin{bmatrix}20&18&-17\\-1&-12&7\\\end{bmatrix}.\\&\text{No row operations, however, will create }\begin{bmatrix}22&9&-7\\2&-9&10\\\end{bmatrix}.\end{align*}$

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Example Question #632 : Linear Algebra

$\begin{align*}&\text{Among the choices given, find the matrix that is not }A=\begin{bmatrix}17&4&-2&7\\-5&14&19&-15\\\end{bmatrix}\\&\text{in an equivalent configuration.}\end{align*}$

Possible Answers:

$\begin{bmatrix}-68&-16&8&-28\\20&-56&-76&60\\\end{bmatrix}$

$\begin{bmatrix}34&8&-4&14\\-30&84&114&-90\\\end{bmatrix}$

$\begin{bmatrix}18&5&-1&8\\-4&15&20&-14\\\end{bmatrix}$

$\begin{bmatrix}22&-10&-21&22\\-5&14&19&-15\\\end{bmatrix}$

Correct answer:

$\begin{bmatrix}18&5&-1&8\\-4&15&20&-14\\\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Two matrices are equivalent if one can be turned into another}\\&\text{via row operations: scaling a row by multiplication of some}\\&\text{constant, addition/subtraction of one row with another, or a}\\&\text{combination of these processes.}\\&\text{The matrix: }\begin{bmatrix}-68&-16&8&-28\\20&-56&-76&60\\\end{bmatrix}\text{ is our entire initial matrix scaled by }-4.\\&\text{The matrix }\begin{bmatrix}34&8&-4&14\\-30&84&114&-90\\\end{bmatrix}\text{ can be formed by scaling the first and second rows of}\\&\text{the orignal by }2\text{ and }6.\\&\text{Finally, subtracting one row from another creates }\begin{bmatrix}22&-10&-21&22\\-5&14&19&-15\\\end{bmatrix}.\\&\text{No row operations, however, will create }\begin{bmatrix}18&5&-1&8\\-4&15&20&-14\\\end{bmatrix}.\end{align*}$

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Example Question #11 : Row Operations And Equivalent Systems

$\begin{align*}&\text{Which of the following matrices is not equivalent to}\\&\begin{bmatrix}2&10&20\\6&6&-11\\\end{bmatrix}?\end{align*}$

Possible Answers:

$\begin{bmatrix}8&40&80\\24&24&-44\\\end{bmatrix}$

$\begin{bmatrix}-\frac{2}{3}&-\frac{10}{3}&-\frac{20}{3}\\\frac{3}{2}&\frac{3}{2}&-\frac{11}{4}\\\end{bmatrix}$

$\begin{bmatrix}-4&4&31\\6&6&-11\\\end{bmatrix}$

$\begin{bmatrix}1&9&19\\5&5&-12\\\end{bmatrix}$

Correct answer:

$\begin{bmatrix}1&9&19\\5&5&-12\\\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Two matrices are equivalent if one can be turned into another}\\&\text{via row operations: scaling a row by multiplication of some}\\&\text{constant, addition/subtraction of one row with another, or a}\\&\text{combination of these processes.}\\&\text{The matrix: }\begin{bmatrix}8&40&80\\24&24&-44\\\end{bmatrix}\text{ is our entire initial matrix scaled by }4.\\&\text{The matrix }\begin{bmatrix}-\frac{2}{3}&-\frac{10}{3}&-\frac{20}{3}\\\frac{3}{2}&\frac{3}{2}&-\frac{11}{4}\\\end{bmatrix}\text{ can be formed by scaling the first and second rows of}\\&\text{the orignal by }-\frac{1}{3}\text{ and }\frac{1}{4}.\\&\text{Finally, subtracting one row from another creates }\begin{bmatrix}-4&4&31\\6&6&-11\\\end{bmatrix}.\\&\text{No row operations, however, will create }\begin{bmatrix}1&9&19\\5&5&-12\\\end{bmatrix}.\end{align*}$

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Example Question #13 : Row Operations And Equivalent Systems

$\begin{align*}&\text{Among the choices given, find the matrix that is not }A=\begin{bmatrix}-13&-15&16&-14\\8&16&-13&3\\\end{bmatrix}\\&\text{in an equivalent configuration.}\end{align*}$

Possible Answers:

$\begin{bmatrix}-\frac{13}{3}&-5&\frac{16}{3}&-\frac{14}{3}\\-32&-64&52&-12\\\end{bmatrix}$

$\begin{bmatrix}-21&-31&29&-17\\8&16&-13&3\\\end{bmatrix}$

$\begin{bmatrix}52&60&-64&56\\-32&-64&52&-12\\\end{bmatrix}$

$\begin{bmatrix}-16&-18&13&-17\\5&13&-16&0\\\end{bmatrix}$

Correct answer:

$\begin{bmatrix}-16&-18&13&-17\\5&13&-16&0\\\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Two matrices are equivalent if one can be turned into another}\\&\text{via row operations: scaling a row by multiplication of some}\\&\text{constant, addition/subtraction of one row with another, or a}\\&\text{combination of these processes.}\\&\text{The matrix: }\begin{bmatrix}52&60&-64&56\\-32&-64&52&-12\\\end{bmatrix}\text{ is our entire initial matrix scaled by }-4.\\&\text{The matrix }\begin{bmatrix}-\frac{13}{3}&-5&\frac{16}{3}&-\frac{14}{3}\\-32&-64&52&-12\\\end{bmatrix}\text{ can be formed by scaling the first and second rows of}\\&\text{the orignal by }\frac{1}{3}\text{ and }-4.\\&\text{Finally, subtracting one row from another creates }\begin{bmatrix}-21&-31&29&-17\\8&16&-13&3\\\end{bmatrix}.\\&\text{No row operations, however, will create }\begin{bmatrix}-16&-18&13&-17\\5&13&-16&0\\\end{bmatrix}.\end{align*}$

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Example Question #71 : Linear Equations

$\begin{align*}&\text{Which of the following matrices is not equivalent to}\\&\begin{bmatrix}-3&-7&-15\\15&-15&18\\\end{bmatrix}?\end{align*}$

Possible Answers:

$\begin{bmatrix}-3&-7&-15\\18&-8&33\\\end{bmatrix}$

$\begin{bmatrix}-2&-6&-14\\16&-14&19\\\end{bmatrix}$

$\begin{bmatrix}9&21&45\\-45&45&-54\\\end{bmatrix}$

$\begin{bmatrix}-1&-\frac{7}{3}&-5\\-60&60&-72\\\end{bmatrix}$

Correct answer:

$\begin{bmatrix}-2&-6&-14\\16&-14&19\\\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Two matrices are equivalent if one can be turned into another}\\&\text{via row operations: scaling a row by multiplication of some}\\&\text{constant, addition/subtraction of one row with another, or a}\\&\text{combination of these processes.}\\&\text{The matrix: }\begin{bmatrix}9&21&45\\-45&45&-54\\\end{bmatrix}\text{ is our entire initial matrix scaled by }-3.\\&\text{The matrix }\begin{bmatrix}-1&-\frac{7}{3}&-5\\-60&60&-72\\\end{bmatrix}\text{ can be formed by scaling the first and second rows of}\\&\text{the orignal by }\frac{1}{3}\text{ and }-4.\\&\text{Finally, subtracting one row from another creates }\begin{bmatrix}-3&-7&-15\\18&-8&33\\\end{bmatrix}.\\&\text{No row operations, however, will create }\begin{bmatrix}-2&-6&-14\\16&-14&19\\\end{bmatrix}.\end{align*}$

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Linear Algebra : Row Operations and Equivalent Systems

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #71 : Linear Equations

Example Question #632 : Linear Algebra

Example Question #11 : Row Operations And Equivalent Systems

Example Question #13 : Row Operations And Equivalent Systems

Example Question #71 : Linear Equations

All Linear Algebra Resources

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