Linear Algebra : Linear Algebra

Study concepts, example questions & explanations for Linear Algebra

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Example Questions

Example Question #681 : Linear Algebra

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

Possible Answers:

\(\displaystyle \text{There is not a unique solution.}\)

\(\displaystyle \text{There is a unique solution.}\)

Correct answer:

\(\displaystyle \text{There is not a unique solution.}\)

Explanation:

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

Example Question #4 : Criteria For Uniqueness And Consistency

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

Possible Answers:

\(\displaystyle \text{There is not a unique solution.}\)

\(\displaystyle \text{There is a unique solution.}\)

Correct answer:

\(\displaystyle \text{There is not a unique solution.}\)

Explanation:

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

Example Question #682 : Linear Algebra

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

Possible Answers:

\(\displaystyle \text{There is not a unique solution.}\)

\(\displaystyle \text{There is a unique solution.}\)

Correct answer:

\(\displaystyle \text{There is a unique solution.}\)

Explanation:

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

Example Question #9 : Criteria For Uniqueness And Consistency

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

Possible Answers:

\(\displaystyle \text{There is a unique solution.}\)

\(\displaystyle \text{There is not a unique solution.}\)

Correct answer:

\(\displaystyle \text{There is not a unique solution.}\)

Explanation:

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

Example Question #683 : Linear Algebra

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

Possible Answers:

\(\displaystyle \text{There is not a unique solution.}\)

\(\displaystyle \text{There is a unique solution.}\)

Correct answer:

\(\displaystyle \text{There is a unique solution.}\)

Explanation:

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

Example Question #684 : Linear Algebra

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

Possible Answers:

\(\displaystyle \text{There is not a unique solution.}\)

\(\displaystyle \text{There is a unique solution.}\)

Correct answer:

\(\displaystyle \text{There is a unique solution.}\)

Explanation:

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

Example Question #685 : Linear Algebra

Compute \(\displaystyle A\cdot B\) where, 

\(\displaystyle A=\begin{bmatrix} 3&4 &0 \\ 2&7 &1 \\ 6&5 &7 \end{bmatrix}\)

\(\displaystyle B =\begin{bmatrix} 2&2 &8 \\ 5&7 &0 \\ 6&4 &3 \end{bmatrix}\)

Possible Answers:

Not Possibe

\(\displaystyle \begin{bmatrix} 79&75 &69 \\ 45&57 &19 \\ 26&34 &24 \end{bmatrix}\)

\(\displaystyle \begin{bmatrix} 26&24 &34 \\ 45&57 &19 \\ 69&75 &71 \end{bmatrix}\)

\(\displaystyle \begin{bmatrix} 45&24 &10 \\ 45&57 &19 \\ 14&80 &70 \end{bmatrix}\)

\(\displaystyle \begin{bmatrix} 26&34 &24 \\ 45&57 &19 \\ 79&75 &69 \end{bmatrix}\)

Correct answer:

\(\displaystyle \begin{bmatrix} 26&34 &24 \\ 45&57 &19 \\ 79&75 &69 \end{bmatrix}\)

Explanation:

Since the number of columns in the first matrix equals the number of rows in the second matrix, we know that these matrices can be multiplied together.  To determine the dimensions of the product matrix, we take the number of rows in the first matrix and the number of columns in the second matrix.  For this example, our product matrix will have dimensions of (3x3). The product matrix equals, \(\displaystyle \begin{bmatrix} (3\cdot 2+4\cdot 5+0\cdot 6)&(3\cdot 2 +4\cdot 7+0\cdot 4)&(3\cdot 8 +4\cdot 0+0\cdot 3) \\ (2\cdot 2 +7\cdot 5+1\cdot 6)&(2\cdot 2 +7\cdot 7+1\cdot 4)&(2\cdot 8 +7\cdot 0+1\cdot 3) \\ (6\cdot 2 +5\cdot 5+7\cdot 6)& (6\cdot 2 +5\cdot 7+7\cdot 4)&(7\cdot 8 +5\cdot 0+7\cdot 3) \end{bmatrix}\)

Example Question #1 : Matrices

Compute \(\displaystyle A\cdot B\) where, 

\(\displaystyle A=\begin{bmatrix} 4&5 &2 \end{bmatrix}\)

\(\displaystyle B=\begin{bmatrix} 0\\ 1 \\ 3 \end{bmatrix}\)

Possible Answers:

\(\displaystyle \begin{bmatrix} 4&5 &6 \end{bmatrix}\)

\(\displaystyle \begin{bmatrix} 11 \end{bmatrix}\)

\(\displaystyle Not Possible\)

\(\displaystyle \begin{bmatrix} 4\\ 5 \\ 6 \end{bmatrix}\)

\(\displaystyle \begin{bmatrix} 4&6 & 5 \end{bmatrix}\)

Correct answer:

\(\displaystyle \begin{bmatrix} 11 \end{bmatrix}\)

Explanation:

Since the number of columns in the first matrix equals the number of rows in the second matrix, we know that these matrices can be multiplied together.  To determine the dimensions of the product matrix, we take the number of rows in the first matrix and the number of columns in the second matrix.  For this example, our product matrix will have dimensions of (1x1). \(\displaystyle \begin{bmatrix} (4\cdot 0+5\cdot 1+3\cdot 2) \end{bmatrix}\)

Example Question #3 : Matrix Matrix Product

Compute \(\displaystyle A\cdot B\) where, 

\(\displaystyle A=\begin{bmatrix} 4&5 & 6 \end{bmatrix}\)

\(\displaystyle B=\begin{bmatrix} 0&1 & 4 \end{bmatrix}\)

Possible Answers:

\(\displaystyle \begin{bmatrix} 4& 8\\ 8&7 \end{bmatrix}\)

\(\displaystyle \begin{bmatrix} 0&5 & 24 \end{bmatrix}\)

Not Possible

\(\displaystyle \begin{bmatrix} 29 \end{bmatrix}\)

\(\displaystyle \begin{bmatrix} 0\\ 5 \\ 24 \end{bmatrix}\)

Correct answer:

Not Possible

Explanation:

In order to be able to multiply matrices, the number of columns of the 1st matrix must equal the number of rows in the second matrix.  Here, the first matrix has dimensions of (1x3). This means it has one row and three columns.  The second matrix has dimensions of (1x3), also one row and three columns.  Since \(\displaystyle \tiny \tiny \small 3\neq1\), we cannot multiply these two matrices together

Example Question #2 : Matrices

Compute \(\displaystyle A\cdot B\), where  

 \(\displaystyle A = \begin{bmatrix} 3&4 \\ 2&7 \\ 6&5 \end{bmatrix}\) 

 \(\displaystyle B = \begin{bmatrix} 2&2 \\ 5&7 \end{bmatrix}\)

Possible Answers:

\(\displaystyle \begin{bmatrix} 34&26 \\ 53&39 \\ 47&37 \end{bmatrix}\)

\(\displaystyle \begin{bmatrix} -14&-22 \\ -31&-45 \\ -13&-23 \end{bmatrix}\)

Not Possible

\(\displaystyle \begin{bmatrix} 6&8 \\ 4&14 \\ 12&10 \end{bmatrix}\)

 \(\displaystyle \begin{bmatrix} 26&34 \\ 39&53 \\ 37&47 \end{bmatrix}\)

Correct answer:

 \(\displaystyle \begin{bmatrix} 26&34 \\ 39&53 \\ 37&47 \end{bmatrix}\)

Explanation:

Since the number of columns in the first matrix equals the number of rows in the second matrix, we know that these matrices can be multiplied together.  To determine the dimensions of the product matrix, we take the number of rows in the first matrix and the number of columns in the second matrix.  For this example, our product matrix will have dimensions of (3x2). The product matrix equals, \(\displaystyle \begin{bmatrix} (3\cdot 2+4\cdot 5)&(3\cdot 2 +4\cdot 7) \\ (2\cdot 2 +7\cdot 5)&(2\cdot 2 +7\cdot 7) \\ (6\cdot 2 +5\cdot 5)& (6\cdot 2 +5\cdot 7) \end{bmatrix}\)

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