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Linear Algebra : Criteria for Uniqueness and Consistency

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Example Question #1 :Criteria For Uniqueness And Consistency

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

Possible Answers:

$\text{There is a unique solution.}$

$\text{There is not a unique solution.}$

Correct answer:

$\text{There is not a unique solution.}$

Explanation:

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

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Example Question #2 :Criteria For Uniqueness And Consistency

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

Possible Answers:

$\text{There is a unique solution.}$

$\text{There is not a unique solution.}$

Correct answer:

$\text{There is a unique solution.}$

Explanation:

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

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Example Question #3 :Criteria For Uniqueness And Consistency

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

Possible Answers:

$\text{There is not a unique solution.}$

$\text{There is a unique solution.}$

Correct answer:

$\text{There is a unique solution.}$

Explanation:

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

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Example Question #4 :Criteria For Uniqueness And Consistency

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

Possible Answers:

$\text{There is a unique solution.}$

$\text{There is not a unique solution.}$

Correct answer:

$\text{There is not a unique solution.}$

Explanation:

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

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Example Question #5 :Criteria For Uniqueness And Consistency

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

Possible Answers:

$\text{There is not a unique solution.}$

$\text{There is a unique solution.}$

Correct answer:

$\text{There is not a unique solution.}$

Explanation:

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

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Example Question #6 :Criteria For Uniqueness And Consistency

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

Possible Answers:

$\text{There is not a unique solution.}$

$\text{There is a unique solution.}$

Correct answer:

$\text{There is a unique solution.}$

Explanation:

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

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Example Question #1 :Criteria For Uniqueness And Consistency

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

Possible Answers:

$\text{There is a unique solution.}$

$\text{There is not a unique solution.}$

Correct answer:

$\text{There is not a unique solution.}$

Explanation:

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

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Example Question #8 :Criteria For Uniqueness And Consistency

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

Possible Answers:

$\text{There is not a unique solution.}$

$\text{There is a unique solution.}$

Correct answer:

$\text{There is not a unique solution.}$

Explanation:

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

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Example Question #9 :Criteria For Uniqueness And Consistency

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

Possible Answers:

$\text{There is not a unique solution.}$

$\text{There is a unique solution.}$

Correct answer:

$\text{There is a unique solution.}$

Explanation:

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

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Example Question #10 :Criteria For Uniqueness And Consistency

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

Possible Answers:

$\text{There is not a unique solution.}$

$\text{There is a unique solution.}$

Correct answer:

$\text{There is not a unique solution.}$

Explanation:

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

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Linear Algebra : Criteria for Uniqueness and Consistency

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

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Example Question #1 :Criteria For Uniqueness And Consistency

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