Linear Equations - Linear Algebra

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Question

$\begin{align}&\text{Decide if there is a unique solution for the equation: }\&\begin{bmatrix}-6&-7&-19\-8&19&-14\0&5&2\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}-21\-36\-149\end{bmatrix}\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b&c \ d&e&f\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)\&det\begin{bmatrix}-6&-7&-19\-8&19&-14\0&5&2\end{bmatrix}=0\&\text{There is not a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Is there a unique solution for the equation }\&\begin{bmatrix}-20&12\7&16\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}-171\149\end{bmatrix}?\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b \ c&d \end{vmatrix}=ad-bc\&det\begin{bmatrix}-20&12\7&16\end{bmatrix}=-404\&\text{There is a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Determine whether or not the equation }\&\begin{bmatrix}6&-8\3&12\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}-188\-101\end{bmatrix}\&\text{has a unique solution.}\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b \ c&d \end{vmatrix}=ad-bc\&det\begin{bmatrix}6&-8\3&12\end{bmatrix}=96\&\text{There is a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Is there a unique solution for the equation }\&\begin{bmatrix}0&-8&2\-5&3&2\20&12&-14\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}-163\36\22\end{bmatrix}?\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b&c \ d&e&f\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)\&det\begin{bmatrix}0&-8&2\-5&3&2\20&12&-14\end{bmatrix}=0\&\text{There is not a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Is there a unique solution for the equation }\&\begin{bmatrix}13&13&-2\12&12&12\-16&-16&-11\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}28\125\107\end{bmatrix}?\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b&c \ d&e&f\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)\&det\begin{bmatrix}13&13&-2\12&12&12\-16&-16&-11\end{bmatrix}=0\&\text{There is not a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Decide if there is a unique solution for the equation: }\&\begin{bmatrix}1&11&14\-1&10&-12\11&-5&16\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}59\4\-32\end{bmatrix}\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b&c \ d&e&f\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)\&det\begin{bmatrix}1&11&14\-1&10&-12\11&-5&16\end{bmatrix}=-2646\&\text{There is a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Determine whether or not the equation }\&\begin{bmatrix}9&-12\-3&4\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}-57\182\end{bmatrix}\&\text{has a unique solution.}\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b \ c&d \end{vmatrix}=ad-bc\&det\begin{bmatrix}9&-12\-3&4\end{bmatrix}=0\&\text{There is not a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Determine whether or not the equation }\&\begin{bmatrix}16&-11\-17&6\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}-8\153\end{bmatrix}\&\text{has a unique solution.}\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b \ c&d \end{vmatrix}=ad-bc\&det\begin{bmatrix}16&-11\-17&6\end{bmatrix}=-91\&\text{There is a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Determine whether or not the equation }\&\begin{bmatrix}2&-5&-12\8&7&1\-10&-2&11\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}-71\-38\-50\end{bmatrix}\&\text{has a unique solution.}\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b&c \ d&e&f\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)\&det\begin{bmatrix}2&-5&-12\8&7&1\-10&-2&11\end{bmatrix}=0\&\text{There is not a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Decide if there is a unique solution for the equation: }\&\begin{bmatrix}19&-12\9&18\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}-126\-163\end{bmatrix}\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b \ c&d \end{vmatrix}=ad-bc\&det\begin{bmatrix}19&-12\9&18\end{bmatrix}=450\&\text{There is a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Decide if there is a unique solution for the equation: }\&\begin{bmatrix}3&-14&-20\5&0&-3\-6&8&14\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}125\155\-70\end{bmatrix}\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b&c \ d&e&f\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)\&det\begin{bmatrix}3&-14&-20\5&0&-3\-6&8&14\end{bmatrix}=0\&\text{There is not a unique solution.}\end{align}$

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Question

$\begin{align}&\text{Determine whether or not the equation }\&\begin{bmatrix}-16&-4&-9\-13&-13&3\20&-18&9\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}179\-130\-184\end{bmatrix}\&\text{has a unique solution.}\end{align}$

Answer

$\begin{align}&\text{To determine whether or not the system of equations represented}\&\text{by a matrix equation has a unique solution, one should calculate}\&\text{the determinant. If the determinant is a nonzero value, then}\&\text{the system of equations has a unique solution. Otherwise, if}\&\text{the determinant is zero, the system of equation may have no}\&\text{solution, or it may have an infinite number of them. To refresh,}\&\text{to calculate a matrix of the given dimensions we use the formula:}\&det\begin{vmatrix} a&b&c \ d&e&f\g&h&i \end{vmatrix}=a(ei-fh)-b(di-fg)+c(dh-eg)\&det\begin{bmatrix}-16&-4&-9\-13&-13&3\20&-18&9\end{bmatrix}=-4146\&\text{There is a unique solution.}\end{align}$

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Question

Solve the following system by reducing its corresponding matrix.

Answer

The corresponding matrix is

$\begin{pmatrix} 1& 2& 0\ 2& 4& 0 \end{pmatrix}$

We add times row one to row two.

$\begin{pmatrix} 1& 2& 0\ 2+(-2)& 4+(-4)& 0+0 \end{pmatrix}=\begin{pmatrix} 1& 2&0 \ 0&0 &0 \end{pmatrix}$

This yields the solution

or

and is a free variable.

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Question

Solve the system by reducing its corresponding matrix.

Answer

The corresponding matrix is

$\begin{pmatrix} 1& 3& 0\ 4& 12& 0 \end{pmatrix}$

We add times row one to row two.

$\begin{pmatrix} 1& 3& 0\ 4+(-4)& 12+(-12)& 0+0 \end{pmatrix}=\begin{pmatrix} 1& 3&0 \ 0&0 &0 \end{pmatrix}$

This yields the solution

or

and is a free variable.

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Question

$\begin{align}&\text{Find the nontrivial solutions for the system of equations: }\&\begin{bmatrix}-10&2&-11\-20&11&-20\\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}0\0\\end{bmatrix}\end{align}$

Answer

$\begin{align}&\text{When faced with a system of equations like the}\&\text{one represented by }\&\begin{bmatrix}-10&2&-11\-20&11&-20\\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}0\0\\end{bmatrix}\&\text{We can manipulate coefficients to eliminate}\&\text{variables in each row. That is to say, to reduce it to echelon form.}\&\text{Reducing our matrix }\begin{bmatrix}-10&2&-11\-20&11&-20\\end{bmatrix}\&\text{We find the reduced echelon form: }\begin{bmatrix}1&0&\frac{81}{70}\0&1&\frac{2}{7}\\end{bmatrix}\&\text{Since each row is equal to zero, we can therefore relate variables to find}\&\text{nontrivial solutions:}\&x=-\frac{(81z)}{70}\&y=-\frac{(2z)}{7}\end{align}$

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Question

$\begin{align}&\text{Find nonzero solutions for }\begin{bmatrix}x\y\z\end{bmatrix}\&\text{for the system: }\begin{bmatrix}4&7&3\17&6&1\\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}0\0\\end{bmatrix}\end{align}$

Answer

$\begin{align}&\text{When faced with a system of equations like the}\&\text{one represented by }\&\begin{bmatrix}4&7&3\17&6&1\\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}0\0\\end{bmatrix}\&\text{We can manipulate coefficients to eliminate}\&\text{variables in each row. That is to say, to reduce it to echelon form.}\&\text{Reducing our matrix }\begin{bmatrix}4&7&3\17&6&1\\end{bmatrix}\&\text{We find the reduced echelon form: }\begin{bmatrix}1&0&-\frac{11}{95}\0&1&\frac{47}{95}\\end{bmatrix}\&\text{Since each row is equal to zero, we can therefore relate variables to find}\&\text{nontrivial solutions:}\&x=\frac{(11z)}{95}\&y=-\frac{(47z)}{95}\end{align}$

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Question

$\begin{align}&\text{Find nontrivial solutions of }\begin{bmatrix}x\y\z\end{bmatrix}\&\text{for the function: }\begin{bmatrix}-14&0&3\-13&-7&9\\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}0\0\\end{bmatrix}\end{align}$

$\begin{align}&x=\frac{(3z)}{14}\&y=-\frac{(87z)}{98}\end{align}$

Answer

$\begin{align}&\text{When faced with a system of equations like the}\&\text{one represented by }\&\begin{bmatrix}-14&0&3\-13&-7&9\\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}0\0\\end{bmatrix}\&\text{We can manipulate coefficients to eliminate}\&\text{variables in each row. That is to say, to reduce it to echelon form.}\&\text{Reducing our matrix }\begin{bmatrix}-14&0&3\-13&-7&9\\end{bmatrix}\&\text{We find the reduced echelon form: }\begin{bmatrix}1&0&-\frac{3}{14}\0&1&-\frac{87}{98}\\end{bmatrix}\&\text{Since each row is equal to zero, we can therefore relate variables to find}\&\text{nontrivial solutions:}\&x=\frac{(3z)}{14}\&y=\frac{(87z)}{98}\end{align}$

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Question

$\begin{align}&\text{Find the nontrivial solutions for the system of equations: }\&\begin{bmatrix}-2&13&12\-9&-18&-18\\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}0\0\\end{bmatrix}\end{align}$

Answer

$\begin{align}&\text{When faced with a system of equations like the}\&\text{one represented by }\&\begin{bmatrix}-2&13&12\-9&-18&-18\\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}0\0\\end{bmatrix}\&\text{We can manipulate coefficients to eliminate}\&\text{variables in each row. That is to say, to reduce it to echelon form.}\&\text{Reducing our matrix }\begin{bmatrix}-2&13&12\-9&-18&-18\\end{bmatrix}\&\text{We find the reduced echelon form: }\begin{bmatrix}1&0&\frac{2}{17}\0&1&\frac{16}{17}\\end{bmatrix}\&\text{Since each row is equal to zero, we can therefore relate variables to find}\&\text{nontrivial solutions:}\&x=-\frac{(2z)}{17}\&y=-\frac{(16z)}{17}\end{align}$

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Question

$\begin{align}&\text{Find nonzero solutions for }\begin{bmatrix}x\y\z\end{bmatrix}\&\text{for the system: }\begin{bmatrix}-7&-2&-11\-19&6&-14\\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}0\0\\end{bmatrix}\end{align}$

Answer

$\begin{align}&\text{When faced with a system of equations like the}\&\text{one represented by }\&\begin{bmatrix}-7&-2&-11\-19&6&-14\\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}0\0\\end{bmatrix}\&\text{We can manipulate coefficients to eliminate}\&\text{variables in each row. That is to say, to reduce it to echelon form.}\&\text{Reducing our matrix }\begin{bmatrix}-7&-2&-11\-19&6&-14\\end{bmatrix}\&\text{We find the reduced echelon form: }\begin{bmatrix}1&0&\frac{47}{40}\0&1&\frac{111}{80}\\end{bmatrix}\&\text{Since each row is equal to zero, we can therefore relate variables to find}\&\text{nontrivial solutions:}\&x=-\frac{(47z)}{40}\&y=-\frac{(111z)}{80}\end{align}$

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Question

$\begin{align}&\text{Find the nontrivial solutions for the system of equations: }\&\begin{bmatrix}-1&16&-18\0&5&10\\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}0\0\\end{bmatrix}\end{align}$

Answer

$\begin{align}&\text{When faced with a system of equations like the}\&\text{one represented by }\&\begin{bmatrix}-1&16&-18\0&5&10\\end{bmatrix}\begin{bmatrix}x\y\z\end{bmatrix}=\begin{bmatrix}0\0\\end{bmatrix}\&\text{We can manipulate coefficients to eliminate}\&\text{variables in each row. That is to say, to reduce it to echelon form.}\&\text{Reducing our matrix }\begin{bmatrix}-1&16&-18\0&5&10\\end{bmatrix}\&\text{We find the reduced echelon form: }\begin{bmatrix}1&0&50\0&1&2\\end{bmatrix}\&\text{Since each row is equal to zero, we can therefore relate variables to find}\&\text{nontrivial solutions:}\&x=-50z\&y=-2z\end{align}$

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