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Linear Algebra : Row Operations and Equivalent Systems

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Example Questions

Example Question #1 : Row Operations And Equivalent Systems

Which matrix is equivalent to

$\begin{pmatrix} 1& 2&5 &-3 \\ 2&4 &6 &8 \end{pmatrix}$

Possible Answers:

$\begin{pmatrix} 1& 2&5 &-3 \\ 1&2 &1 &11 \end{pmatrix}$

$\begin{pmatrix} 1& 2&5 &3 \\ 2&4 &6 &8 \end{pmatrix}$

$\begin{pmatrix} 3& -2&11 &5 \\ 2&4 &6 &8 \end{pmatrix}$

$\begin{pmatrix} 1& 2&5 &-3 \\ 3&6 &11 &11 \end{pmatrix}$

Correct answer:

$\begin{pmatrix} 1& 2&5 &-3 \\ 1&2 &1 &11 \end{pmatrix}$

Explanation:

By adding times row one to row two we get

$\begin{pmatrix} 1& 2&5 &-3 \\ 2+(-1)&4+(-2) &6+(-5) &8+3 \end{pmatrix}$

$=\begin{pmatrix} 1& 2&5 &-3 \\ 1&2 &1 &11 \end{pmatrix}$

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Example Question #1 : Row Operations And Equivalent Systems

Which matrix is equivalent to

$\begin{pmatrix} 3&6 \\ 5&7 \end{pmatrix}$

Possible Answers:

$\begin{pmatrix} -2&1 \\ 5&7 \end{pmatrix}$

$\begin{pmatrix} 3&6 \\ 8&1 \end{pmatrix}$

$\begin{pmatrix} 5&13 \\ 4&2 \end{pmatrix}$

$\begin{pmatrix} 3&6 \\ 8&13 \end{pmatrix}$

Correct answer:

$\begin{pmatrix} 3&6 \\ 8&13 \end{pmatrix}$

Explanation:

By adding row one to row two we get

$\begin{pmatrix} 3&6 \\ 5+3&7+6 \end{pmatrix}$

$=\begin{pmatrix} 3&6 \\ 8&13 \end{pmatrix}$

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Example Question #2 : Row Operations And Equivalent Systems

Which matrix is equivalent to

$\begin{pmatrix} 3&6 \\ 9&4 \end{pmatrix}$

Possible Answers:

$\begin{pmatrix} 4&6 \\ 9&3 \end{pmatrix}$

$\begin{pmatrix} 4&9 \\ 6&3 \end{pmatrix}$

$\begin{pmatrix} 3&9 \\ 6&4 \end{pmatrix}$

$\begin{pmatrix} 12&10 \\ 9&4 \end{pmatrix}$

Correct answer:

$\begin{pmatrix} 12&10 \\ 9&4 \end{pmatrix}$

Explanation:

We add row two to row one to get

$\begin{pmatrix} 3+9&6+4 \\ 9&4 \end{pmatrix}=\begin{pmatrix} 12&10 \\ 9&4 \end{pmatrix}$

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Example Question #3 : Row Operations And Equivalent Systems

$\begin{align*}&\text{Among the choices given, find the matrix that is not }A=\begin{bmatrix}-6&-12&18&7\\-19&-10&-11&-11\\\end{bmatrix}\\&\text{in an equivalent configuration.}\end{align*}$

Possible Answers:

$\begin{bmatrix}-9&-15&15&4\\-22&-13&-14&-14\\\end{bmatrix}$

$\begin{bmatrix}-\frac{3}{2}&-3&\frac{9}{2}&\frac{7}{4}\\-\frac{19}{4}&-\frac{5}{2}&-\frac{11}{4}&-\frac{11}{4}\\\end{bmatrix}$

$\begin{bmatrix}-6&-12&18&7\\-13&2&-29&-18\\\end{bmatrix}$

$\begin{bmatrix}-12&-24&36&14\\-95&-50&-55&-55\\\end{bmatrix}$

Correct answer:

$\begin{bmatrix}-9&-15&15&4\\-22&-13&-14&-14\\\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Two matrices are equivalent if one can be turned into another}\\&\text{via row operations: scaling a row by multiplication of some}\\&\text{constant, addition/subtraction of one row with another, or a}\\&\text{combination of these processes..}\\&\text{The matrix: }\begin{bmatrix}-\frac{3}{2}&-3&\frac{9}{2}&\frac{7}{4}\\-\frac{19}{4}&-\frac{5}{2}&-\frac{11}{4}&-\frac{11}{4}\\\end{bmatrix}\text{ is our entire initial matrix scaled by }\frac{1}{4}\\&\text{The matrix }\begin{bmatrix}-12&-24&36&14\\-95&-50&-55&-55\\\end{bmatrix}\text{ can be formed by scaling the first and second rows of}\\&\text{the orignal by }2\text{ and }5\\&\text{Finally, subtracting one row from another creates }\begin{bmatrix}-6&-12&18&7\\-13&2&-29&-18\\\end{bmatrix}\\&\text{No row operations, however, will create }\begin{bmatrix}-9&-15&15&4\\-22&-13&-14&-14\\\end{bmatrix}\end{align*}$

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Example Question #4 : Row Operations And Equivalent Systems

$\begin{align*}&\text{Which of the following matrices is not equivalent to}\\&\begin{bmatrix}4&14&-11&18\\-8&-15&7&12\\\end{bmatrix}?\end{align*}$

Possible Answers:

$\begin{bmatrix}2&7&-\frac{11}{2}&9\\-4&-\frac{15}{2}&\frac{7}{2}&6\\\end{bmatrix}$

$\begin{bmatrix}12&29&-18&6\\-8&-15&7&12\\\end{bmatrix}$

$\begin{bmatrix}2&7&-\frac{11}{2}&9\\48&90&-42&-72\\\end{bmatrix}$

$\begin{bmatrix}7&17&-8&21\\-5&-12&10&15\\\end{bmatrix}$

Correct answer:

$\begin{bmatrix}7&17&-8&21\\-5&-12&10&15\\\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Two matrices are equivalent if one can be turned into another}\\&\text{via row operations: scaling a row by multiplication of some}\\&\text{constant, addition/subtraction of one row with another, or a}\\&\text{combination of these processes..}\\&\text{The matrix: }\begin{bmatrix}2&7&-\frac{11}{2}&9\\-4&-\frac{15}{2}&\frac{7}{2}&6\\\end{bmatrix}\text{ is our entire initial matrix scaled by }\frac{1}{2}.\\&\text{The matrix }\begin{bmatrix}2&7&-\frac{11}{2}&9\\48&90&-42&-72\\\end{bmatrix}\text{ can be formed by scaling the first and second rows of}\\&\text{the orignal by }\frac{1}{2}\text{ and }-6.\\&\text{Finally, subtracting one row from another creates }\begin{bmatrix}12&29&-18&6\\-8&-15&7&12\\\end{bmatrix}.\\&\text{No row operations, however, will create }\begin{bmatrix}7&17&-8&21\\-5&-12&10&15\\\end{bmatrix}.\end{align*}$

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Example Question #5 : Row Operations And Equivalent Systems

$\begin{align*}&\text{Identify the matrix which is not a equivalent to:}\\&\begin{bmatrix}6&17&18\\16&-17&8\\\end{bmatrix}\end{align*}$

Possible Answers:

$\begin{bmatrix}6&17&18\\10&-34&-10\\\end{bmatrix}$

$\begin{bmatrix}\frac{6}{5}&\frac{17}{5}&\frac{18}{5}\\\frac{16}{5}&-\frac{17}{5}&\frac{8}{5}\\\end{bmatrix}$

$\begin{bmatrix}-18&-51&-54\\-96&102&-48\\\end{bmatrix}$

$\begin{bmatrix}3&14&15\\13&-20&5\\\end{bmatrix}$

Correct answer:

$\begin{bmatrix}3&14&15\\13&-20&5\\\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Two matrices are equivalent if one can be turned into another}\\&\text{via row operations: scaling a row by multiplication of some}\\&\text{constant, addition/subtraction of one row with another, or a}\\&\text{combination of these processes..}\\&\text{The matrix: }\begin{bmatrix}\frac{6}{5}&\frac{17}{5}&\frac{18}{5}\\\frac{16}{5}&-\frac{17}{5}&\frac{8}{5}\\\end{bmatrix}\text{ is our entire initial matrix scaled by }\frac{1}{5}.\\&\text{The matrix }\begin{bmatrix}-18&-51&-54\\-96&102&-48\\\end{bmatrix}\text{ can be formed by scaling the first and second rows of}\\&\text{the orignal by }-3\text{ and }-6.\\&\text{Finally, subtracting one row from another creates }\begin{bmatrix}6&17&18\\10&-34&-10\\\end{bmatrix}.\\&\text{No row operations, however, will create }\begin{bmatrix}3&14&15\\13&-20&5\\\end{bmatrix}.\end{align*}$

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Example Question #6 : Row Operations And Equivalent Systems

$\begin{align*}&\text{Which of the following matrices is not equivalent to}\\&\begin{bmatrix}4&-15&16\\11&3&14\\\end{bmatrix}?\end{align*}$

Possible Answers:

$\begin{bmatrix}8&-30&32\\22&6&28\\\end{bmatrix}$

$\begin{bmatrix}4&-15&16\\7&18&-2\\\end{bmatrix}$

$\begin{bmatrix}-2&\frac{15}{2}&-8\\-\frac{11}{6}&-\frac{1}{2}&-\frac{7}{3}\\\end{bmatrix}$

$\begin{bmatrix}1&-18&13\\8&0&11\\\end{bmatrix}$

Correct answer:

$\begin{bmatrix}1&-18&13\\8&0&11\\\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Two matrices are equivalent if one can be turned into another}\\&\text{via row operations: scaling a row by multiplication of some}\\&\text{constant, addition/subtraction of one row with another, or a}\\&\text{combination of these processes..}\\&\text{The matrix: }\begin{bmatrix}8&-30&32\\22&6&28\\\end{bmatrix}\text{ is our entire initial matrix scaled by }2.\\&\text{The matrix }\begin{bmatrix}-2&\frac{15}{2}&-8\\-\frac{11}{6}&-\frac{1}{2}&-\frac{7}{3}\\\end{bmatrix}\text{ can be formed by scaling the first and second rows of}\\&\text{the orignal by }-\frac{1}{2}\text{ and }-\frac{1}{6}.\\&\text{Finally, subtracting one row from another creates }\begin{bmatrix}4&-15&16\\7&18&-2\\\end{bmatrix}.\\&\text{No row operations, however, will create }\begin{bmatrix}1&-18&13\\8&0&11\\\end{bmatrix}.\end{align*}$

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Example Question #7 : Row Operations And Equivalent Systems

$\begin{align*}&\text{Among the choices given, find the matrix that is not }A=\begin{bmatrix}-13&11&17&-10\\-12&-15&-5&-2\\\end{bmatrix}\\&\text{in an equivalent configuration.}\end{align*}$

Possible Answers:

$\begin{bmatrix}\frac{13}{4}&-\frac{11}{4}&-\frac{17}{4}&\frac{5}{2}\\3&\frac{15}{4}&\frac{5}{4}&\frac{1}{2}\\\end{bmatrix}$

$\begin{bmatrix}\frac{13}{3}&-\frac{11}{3}&-\frac{17}{3}&\frac{10}{3}\\-2&-\frac{5}{2}&-\frac{5}{6}&-\frac{1}{3}\\\end{bmatrix}$

$\begin{bmatrix}-1&26&22&-8\\-12&-15&-5&-2\\\end{bmatrix}$

$\begin{bmatrix}-14&10&16&-11\\-13&-16&-6&-3\\\end{bmatrix}$

Correct answer:

$\begin{bmatrix}-14&10&16&-11\\-13&-16&-6&-3\\\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Two matrices are equivalent if one can be turned into another}\\&\text{via row operations: scaling a row by multiplication of some}\\&\text{constant, addition/subtraction of one row with another, or a}\\&\text{combination of these processes..}\\&\text{The matrix: }\begin{bmatrix}\frac{13}{4}&-\frac{11}{4}&-\frac{17}{4}&\frac{5}{2}\\3&\frac{15}{4}&\frac{5}{4}&\frac{1}{2}\\\end{bmatrix}\text{ is our entire initial matrix scaled by }-\frac{1}{4}.\\&\text{The matrix }\begin{bmatrix}\frac{13}{3}&-\frac{11}{3}&-\frac{17}{3}&\frac{10}{3}\\-2&-\frac{5}{2}&-\frac{5}{6}&-\frac{1}{3}\\\end{bmatrix}\text{ can be formed by scaling the first and second rows of}\\&\text{the orignal by }-\frac{1}{3}\text{ and }\frac{1}{6}.\\&\text{Finally, subtracting one row from another creates }\begin{bmatrix}-1&26&22&-8\\-12&-15&-5&-2\\\end{bmatrix}.\\&\text{No row operations, however, will create }\begin{bmatrix}-14&10&16&-11\\-13&-16&-6&-3\\\end{bmatrix}.\end{align*}$

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Example Question #8 : Row Operations And Equivalent Systems

$\begin{align*}&\text{Which of the following matrices is not equivalent to}\\&\begin{bmatrix}-12&-1&8\\10&3&3\\\end{bmatrix}?\end{align*}$

Possible Answers:

$\begin{bmatrix}-4&-\frac{1}{3}&\frac{8}{3}\\50&15&15\\\end{bmatrix}$

$\begin{bmatrix}-14&-3&6\\8&1&1\\\end{bmatrix}$

$\begin{bmatrix}-6&-\frac{1}{2}&4\\5&\frac{3}{2}&\frac{3}{2}\\\end{bmatrix}$

$\begin{bmatrix}-12&-1&8\\22&4&-5\\\end{bmatrix}$

Correct answer:

$\begin{bmatrix}-14&-3&6\\8&1&1\\\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Two matrices are equivalent if one can be turned into another}\\&\text{via row operations: scaling a row by multiplication of some}\\&\text{constant, addition/subtraction of one row with another, or a}\\&\text{combination of these processes..}\\&\text{The matrix: }\begin{bmatrix}-6&-\frac{1}{2}&4\\5&\frac{3}{2}&\frac{3}{2}\\\end{bmatrix}\text{ is our entire initial matrix scaled by }\frac{1}{2}.\\&\text{The matrix }\begin{bmatrix}-4&-\frac{1}{3}&\frac{8}{3}\\50&15&15\\\end{bmatrix}\text{ can be formed by scaling the first and second rows of}\\&\text{the orignal by }\frac{1}{3}\text{ and }5.\\&\text{Finally, subtracting one row from another creates }\begin{bmatrix}-12&-1&8\\22&4&-5\\\end{bmatrix}.\\&\text{No row operations, however, will create }\begin{bmatrix}-14&-3&6\\8&1&1\\\end{bmatrix}.\end{align*}$

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Example Question #3 : Row Operations And Equivalent Systems

$\begin{align*}&\text{Which of the following matrices is not equivalent to}\\&\begin{bmatrix}-12&9&19\\3&-4&12\\\end{bmatrix}?\end{align*}$

Possible Answers:

$\begin{bmatrix}6&-\frac{9}{2}&-\frac{19}{2}\\-\frac{1}{2}&\frac{2}{3}&-2\\\end{bmatrix}$

$\begin{bmatrix}-15&13&7\\3&-4&12\\\end{bmatrix}$

$\begin{bmatrix}-10&11&21\\5&-2&14\\\end{bmatrix}$

$\begin{bmatrix}\frac{12}{5}&-\frac{9}{5}&-\frac{19}{5}\\-\frac{3}{5}&\frac{4}{5}&-\frac{12}{5}\\\end{bmatrix}$

Correct answer:

$\begin{bmatrix}-10&11&21\\5&-2&14\\\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Two matrices are equivalent if one can be turned into another}\\&\text{via row operations: scaling a row by multiplication of some}\\&\text{constant, addition/subtraction of one row with another, or a}\\&\text{combination of these processes..}\\&\text{The matrix: }\begin{bmatrix}\frac{12}{5}&-\frac{9}{5}&-\frac{19}{5}\\-\frac{3}{5}&\frac{4}{5}&-\frac{12}{5}\\\end{bmatrix}\text{ is our entire initial matrix scaled by }-\frac{1}{5}.\\&\text{The matrix }\begin{bmatrix}6&-\frac{9}{2}&-\frac{19}{2}\\-\frac{1}{2}&\frac{2}{3}&-2\\\end{bmatrix}\text{ can be formed by scaling the first and second rows of}\\&\text{the orignal by }-\frac{1}{2}\text{ and }-\frac{1}{6}.\\&\text{Finally, subtracting one row from another creates }\begin{bmatrix}-15&13&7\\3&-4&12\\\end{bmatrix}.\\&\text{No row operations, however, will create }\begin{bmatrix}-10&11&21\\5&-2&14\\\end{bmatrix}.\end{align*}$

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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 #1 : Row Operations And Equivalent Systems

Example Question #1 : Row Operations And Equivalent Systems

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Example Question #3 : Row Operations And Equivalent Systems

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Example Question #5 : Row Operations And Equivalent Systems

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