Row Operations and Equivalent Systems - Linear Algebra

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Question

Which matrix is equivalent to

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

Answer

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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Question

Which matrix is equivalent to

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

Answer

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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Question

Which matrix is equivalent to

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

Answer

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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Question

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

Answer

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

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Question

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

Answer

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

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Question

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

Answer

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

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Question

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

Answer

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

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Question

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

Answer

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

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Question

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

Answer

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

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Question

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

Answer

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

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Question

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

Answer

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

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Question

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

Answer

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

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Question

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

Answer

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

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Question

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

Answer

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

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Question

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

Answer

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

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