Row Operations and Equivalent Systems - Linear Algebra

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Question

Which matrix is equivalent to

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

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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

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

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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

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

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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.}$\end{align}$

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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}$\&\begin{bmatrix}4&14&-11&18\-8&-15&7&12\\end{bmatrix}?\end{align}$

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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:}$\&\begin{bmatrix}6&17&18\16&-17&8\\end{bmatrix}\end{align}$

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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}$\&\begin{bmatrix}4&-15&16\11&3&14\\end{bmatrix}?\end{align}$

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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.}$\end{align}$

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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}$\&\begin{bmatrix}-12&-1&8\10&3&3\\end{bmatrix}?\end{align}$

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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}$\&\begin{bmatrix}-12&9&19\3&-4&12\\end{bmatrix}?\end{align}$

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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:}$\&\begin{bmatrix}19&6&-10\-1&-12&7\\end{bmatrix}\end{align}$

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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.}$\end{align}$

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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}$\&\begin{bmatrix}2&10&20\6&6&-11\\end{bmatrix}?\end{align}$

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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.}$\end{align}$

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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}$\&\begin{bmatrix}-3&-7&-15\15&-15&18\\end{bmatrix}?\end{align}$

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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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Question

Which matrix is equivalent to

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

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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

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

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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

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

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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.}$\end{align}$

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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}$\&\begin{bmatrix}4&14&-11&18\-8&-15&7&12\\end{bmatrix}?\end{align}$

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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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