Determinants_ Properties and Applications_ Attempt review _ eClass

3/27/24, 1:25 AM Determinants: Properties and Applications: Attempt review | eClass https://eclass.srv.ualberta.ca/mod/quiz/review.php?attempt=15149458&cmid=7601901 1/22 Started on Sunday, 17 March 2024, 12:58 AM State Finished Completed on Monday, 18 March 2024, 5:57 PM Time taken 1 day 16 hours Grade 101.00 out of 101.00 ( 100 %)

3/27/24, 1:25 AM Determinants: Properties and Applications: Attempt review | eClass https://eclass.srv.ualberta.ca/mod/quiz/review.php?attempt=15149458&cmid=7601901 2/22 Question 1 Correct Mark 5.00 out of 5.00 Properties of the Determinant Many of the properties below are discussed in the video Determinants: Properties and Applications 1 . You should know these properties. Many of these have a simple explanation, but for others the proof is more complicated, so most of these will be skipped. The video provides a partial proof of property (c) below. Theorem: Let be an matrix. (a) For any scalar , (b) For any matrix , (c) is invertible if and only if 0  . Furthermore, if is invertible, then (d) For the transpose of , we have (e) If contains a row or column of zeros, or if contains a row (resp. column) which is a scalar multiple of another row (resp.column) then 0  (f) If and are matrices such that , , and are identical except that the row (resp. column) of is the sum of the rows (resp. columns) of and , that is, then Proof: We will prove . Assume: is invertible, so that  . ≠ 0

3/27/24, 1:25 AM Determinants: Properties and Applications: Attempt review | eClass https://eclass.srv.ualberta.ca/mod/quiz/review.php?attempt=15149458&cmid=7601901 4/22 Question 2 Correct Mark 7.00 out of 7.00 Properties of the Determinant: Examples Example: Let . By Property  , . Let's compute directly to get a sense of why this property holds: 2  4  8  Example: Use Property (c) to determine whether the following matrix is invertible: Since -9   , the matrix  invertible. (a) ≠ 0 is Correct Marks for this submission: 7.00/7.00.

3/27/24, 1:25 AM Determinants: Properties and Applications: Attempt review | eClass https://eclass.srv.ualberta.ca/mod/quiz/review.php?attempt=15149458&cmid=7601901 5/22 Question 3 Correct Mark 8.00 out of 8.00 Example: Let . Since column  of is scalar multiple of column 1, we know by Property  that 0  . Indeed, 0  (by cofactor expansion along column 3) Recall Property (f): If , , and are matrices which are identical except that then Example: Let . Note: Thus by Property  0  24  (see example directly above) 24  . 3 (e) (f) Correct

3/27/24, 1:25 AM Determinants: Properties and Applications: Attempt review | eClass https://eclass.srv.ualberta.ca/mod/quiz/review.php?attempt=15149458&cmid=7601901 7/22 Question 5 Correct Mark 8.00 out of 8.00 Example: Is , for all matrices and ? Correct answer, well done. Marks for this submission: 1.00/1.00. Counterexample: Take and , where is the identity matrix. Thus 0 0 0 0 Correct answer, well done. Marks for this submission: 1.00/1.00. Then 0 ,but 1 1 2 Correct answer, well done. Marks for this submission: 1.00/1.00. Correct answer, well done. Marks for this submission: 1.00/1.00. Correct answer, well done. Marks for this submission: 1.00/1.00. Correct answer, well done. Marks for this submission: 1.00/1.00. Thus in this case, Correct answer, well done. Marks for this submission: 1.00/1.00. and hence in general. Correct answer, well done. Marks for this submission: 1.00/1.00. No ≠ ≠ A correct answer is: "No" A correct answer is . A correct answer is , which can be typed in as follows: 0 A correct answer is , which can be typed in as follows: 1 A correct answer is , which can be typed in as follows: 1 A correct answer is , which can be typed in as follows: 2 A correct answer is: "≠" A correct answer is: "≠"

3/27/24, 1:25 AM Determinants: Properties and Applications: Attempt review | eClass https://eclass.srv.ualberta.ca/mod/quiz/review.php?attempt=15149458&cmid=7601901 8/22 Question 6 Correct Mark 2.00 out of 2.00 Applications of the Determinant We have already discussed how the determinant of a square matrix \(A\) can be used to determine if \(A\) is invertible. Indeed, \(A\) is invertible if and only if \(\det(A) \)  . This, in turn, gives us an application to linear transformations: a linear transformation \(T: \mathbb{R}^n \rightarrow \mathbb{R}^n\) is invertible if and only if the standard matrix \([T]\) of \(T\) is invertible if and only if \(\det([T]) \)  . Determinants are useful in other ways. In this learning activity, we will discuss how determinants can be used to compute the inverse of an invertible matrix, as well as some applications of determinants to geometry, including the concept of the cross product of two vectors in \ (\mathbb{R}^3\). Later in Block 6, we will use determinants to compute what is known as the characteristic polynomial of a square matrix. ≠ 0 ≠ 0 Correct Marks for this submission: 2.00/2.00.

3/27/24, 1:25 AM Determinants: Properties and Applications: Attempt review | eClass https://eclass.srv.ualberta.ca/mod/quiz/review.php?attempt=15149458&cmid=7601901 10/22 A correct answer is \( a \), which can be typed in as follows: a A correct answer is \( \left[\begin{array}{cc} d & -b \\ -c & a \end{array}\right] \).

3/27/24, 1:25 AM Determinants: Properties and Applications: Attempt review | eClass https://eclass.srv.ualberta.ca/mod/quiz/review.php?attempt=15149458&cmid=7601901 11/22 Question 8 Correct Mark 6.00 out of 6.00 Example: Use the adjoint formula for the inverse of a matrix to find the inverse of the invertible matrix \[A = {\left[\begin{array}{ccc} 2 & 1 & 4 \\ 1 & 0 & -1 \\ 0 & 1 & 8 \end{array}\right]}\] Let's start by computing the adjoint of \(A\). Note: in the second line of this computation, "\(+\)" and "\( - \)" represent the sign of the cofactors \(C_{ij} = (-1)^{i+j}\det(A_{ij})\), which are determined by the term \((-1)^{i+j}\). \ (\operatorname{adj} (A)\) \ (=\) \( \left[\begin{array}{cccc}C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23} \\ C_{31} & C_{32} & C_{33} \end{array} \right]^{ \ T}\) \ (=\) \( \left[\begin{array}{cccccc} +\begin{vmatrix}0&-1\\1&8\end{vmatrix}&& - \begin{vmatrix}1&-1\\0&8\end{vmatrix} && +\begin{vmatrix}1&0\\0&1\end{vmatrix}\; \; \\ - \begin{vmatrix}1&4\\1&8\end{vmatrix}& &+\begin{vmatrix}2&4\\0&8\end{vmatrix}& & - \begin{vmatrix}2&1\\0&1\end{vmatrix}\; \; \\ + \begin{vmatrix}1&4\\0&-1\end{vmatrix} && - \begin{vmatrix}2&4\\1&-1\end{vmatrix} && + \begin{vmatrix}2&1\\1&0\end{vmatrix} \; \; \end{array} \right]^{\ T}\) \ (=\) \( \left( \begin{array}{c}\hspace{0.1cm}\\ \\ \\ \\\end{array}\right.\) 1 -8 1 -4 16 -2 -1 6 -1 \ (\left. \begin{array}{c}\hspace{0.1cm}\\ \\ \\ \\\end{array}\right)^T\) Correct answer, well done. Marks for this submission: 3.00/3.00. \ (=\) 1 -4 -1 -8 16 6 1 -2 -1 Correct answer, well done. Marks for this submission: 1.00/1.00. Now, we will compute the determinant of \(A\): \(\det(A) = \) -2 Correct answer, well done. Marks for this submission: 1.00/1.00. Therefore, the inverse of \(A\) is

3/27/24, 1:25 AM Determinants: Properties and Applications: Attempt review | eClass https://eclass.srv.ualberta.ca/mod/quiz/review.php?attempt=15149458&cmid=7601901 13/22 Question 9 Correct Mark 13.00 out of 13.00 Cross Product Definition: If \(\mathbf{u}= [u_1, u_2, u_3]\) and \(\mathbf{v}= [v_1, v_2, v_3]\) are vectors in \(\mathbb{R}^3\), then the cross product of \ (\mathbf{u}\) and \(\mathbf{v}\) is the vector in \(\mathbb{R}^3\) given by \[\mathbf{u}\times\mathbf{v}= [u_2v_3−u_3v_2,\, u_3v_1−u_1v_3,\, u_1v_2−u_2v_1]\] In determinant notation \[\begin{aligned}\mathbf{u}\times\mathbf{v} &= \left[ \, \begin{vmatrix} u_2 & u_3\\ v_2 & v_3 \end{vmatrix}, -\begin{vmatrix} u_1 & u_3\\ v_1 & v_3 \end{vmatrix}, \begin{vmatrix} u_1 & u_2\\ v_1 & v_2 \end{vmatrix} \, \right] \end{aligned}\] As explained later, the cross product \( \mathbf{u}\times\mathbf{v}\) is orthogonal to both \( \mathbf{u}\) and \( \mathbf{v}\). It has applications in mathematics and physics. Let \(\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3\) be the standard basis vectors in \(\mathbb{R}^3\). A convenient way to remember this formula is to view it as the following determinant \[\begin{aligned} \mathbf{u} \times \mathbf{v} &=\left| \begin{array}{ccc} \mathbf{e}_1 & \mathbf{e}_2 & \mathbf{e}_3 \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{array} \right| \\ \\&= \begin{vmatrix} u_2 & u_3\\ v_2 & v_3 \end{vmatrix}\mathbf{e}_1 - \begin{vmatrix} u_1 & u_3\\ v_1 & v_3 \end{vmatrix} \mathbf{e}_2 + \begin{vmatrix} u_1 & u_2\\ v_1 & v_2 \end{vmatrix} \mathbf{e}_3 \\ \\ &= \left[ \, \begin{vmatrix} u_2 & u_3\\ v_2 & v_3 \end{vmatrix}, -\begin{vmatrix} u_1 & u_3\\ v_1 & v_3 \end{vmatrix}, \begin{vmatrix} u_1 & u_2\\ v_1 & v_2 \end{vmatrix} \, \right], \end{aligned}\] where we must use cofactor expansion along row 1. Viewing the cross product as this determinant not only provides us with a helpful way to remember the formula, but it will also enable us to use the properties of the determinant to understand the properties of the cross product, which will be presented in Question 12. Example: Let \(\mathbf{u} = [3,2,1]\) and \(\mathbf{v} = [-1,1,0]\). Compute \(\mathbf{u} \times \mathbf{v}\). \(\mathbf{u} \times \mathbf{v}\) \(= \det \) 2 1 1 0 \(\mathbf{e}_1 \; - \; \det \) 3 1 -1 0 \(\mathbf{e}_2 \; + \; \det \) 3 2 -1 1 \(\mathbf{e}_3\) Correct answer, well done. Marks for this submission: 1.00/1.00. Correct answer, well done. Marks for this submission: 1.00/1.00. Correct answer, well done. Marks for this submission: 1.00/1.00. \(=\) -1 \(\mathbf{e}_1 \; - \; \) 1 \(\mathbf{e}_2 \; + \; \) 5 \(\mathbf{e}_3\) Correct answer, well done. Marks for this submission: 1.00/1.00. Correct answer, well done. Marks for this submission: 1.00/1.00. Correct answer, well done. Marks for this submission: 1.00/1.00. \(=\) -1 -1 5

3/27/24, 1:25 AM Determinants: Properties and Applications: Attempt review | eClass https://eclass.srv.ualberta.ca/mod/quiz/review.php?attempt=15149458&cmid=7601901 14/22 \(\mathbf{u} \times \mathbf{v}\) Correct answer, well done. Marks for this submission: 1.00/1.00. Exercise: Calculate the following cross products. (a) \([1,2,3]\times[4,3,2]=\) -5 10 -5 Correct answer, well done. Marks for this submission: 2.00/2.00. (b) \([1,1,-1]\times[2,0,2]=\) 2 -4 -2 Correct answer, well done. Marks for this submission: 2.00/2.00. (c) \([2,0,2]\times[1,1,-1]=\) -2 4 2 Correct answer, well done. Marks for this submission: 2.00/2.00. A correct answer is \( \left[\begin{array}{cc} 2 & 1 \\ 1 & 0 \end{array}\right] \). A correct answer is \( \left[\begin{array}{cc} 3 & 1 \\ -1 & 0 \end{array}\right] \). A correct answer is \( \left[\begin{array}{cc} 3 & 2 \\ -1 & 1 \end{array}\right] \). A correct answer is \( -1 \), which can be typed in as follows: -1 A correct answer is \( 1 \), which can be typed in as follows: 1 A correct answer is \( 5 \), which can be typed in as follows: 5 A correct answer is \( \left[\begin{array}{ccc} -1 & -1 & 5 \end{array}\right] \). A correct answer is \( \left[\begin{array}{ccc} -5 & 10 & -5 \end{array}\right] \). A correct answer is \( \left[\begin{array}{ccc} 2 & -4 & -2 \end{array}\right] \). A correct answer is \( \left[\begin{array}{ccc} -2 & 4 & 2 \end{array}\right] \).

3/27/24, 1:25 AM Determinants: Properties and Applications: Attempt review | eClass https://eclass.srv.ualberta.ca/mod/quiz/review.php?attempt=15149458&cmid=7601901 16/22 Question 11 Correct Mark 1.00 out of 1.00 Then \(\mathbf{u}\), \(\mathbf{v}\) and \(\mathbf{w}\) lie in the same plane if and only if \(\mathbf{u}\cdot(\mathbf{v}\times\mathbf{w}) =\) 0  . Example: Do the vectors \(\mathbf{u} = [1,2,1]\), \(\mathbf{v} = [2,0,3]\) and \(\mathbf{w} = [-1,1,0]\) lie in the same plane?  , since \(\mathbf{u}\cdot(\mathbf{v}\times\mathbf{w}) = \) -7   . No ≠ 0 Correct Marks for this submission: 6.00/6.00. Area of a Parallelogram in \(\mathbb{R}^2\) The determinant of a \(2 \times 2\) matrix also has a nice geometric interpretation. Theorem: Let \(\mathbf{u} = [u_1,u_2]\) and \(\mathbf{v} = [v_1, v_2]\) be vectors in \(\mathbb{R}^2\). Then the area \(\mathcal{A}\) of the parallelogram in \(\mathbb{R}^2\) determined by \(\mathbf{u}\) and \(\mathbf{v}\) is given by the absolute value of \(\det \begin{bmatrix}u_1 & u_2\\v_1&v_2\end{bmatrix}\), that is, \[ \mathcal{A} = \left| \, \det \begin{bmatrix}u_1 & u_2\\v_1&v_2\end{bmatrix} \, \right | \] Example: Find the area \(\mathcal{A}\) of the parallelogram in \(\mathbb{R}^2\) determined by the vectors \(\mathbf{u} = [-1,4]\) and \ (\mathbf{v} = [3,-2]\). \( \mathcal{A} = \left| \, \det \begin{bmatrix}-1&4\\3&-2\end{bmatrix} \, \right| =\) 10  Correct Marks for this submission: 1.00/1.00.

3/27/24, 1:25 AM Determinants: Properties and Applications: Attempt review | eClass https://eclass.srv.ualberta.ca/mod/quiz/review.php?attempt=15149458&cmid=7601901 17/22 Question 12 Correct Mark 8.00 out of 8.00 Properties of the Cross Product Recall: \[\begin{aligned} \mathbf{u} \times \mathbf{v} &=\left| \begin{array}{ccc} \mathbf{e}_1 & \mathbf{e}_2 & \mathbf{e}_3 \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{array} \right| \\ \\ &= \left[ \, \begin{vmatrix} u_2 & u_3\\ v_2 & v_3 \end{vmatrix}, -\begin{vmatrix} u_1 & u_3\\ v_1 & v_3 \end{vmatrix}, \begin{vmatrix} u_1 & u_2\\ v_1 & v_2 \end{vmatrix} \, \right], \end{aligned}\] When considering the properties of the cross product below, keep in mind the properties of the determinant. For the following statements, let \(\mathbf{u},\,\mathbf{v}\) and \(\mathbf{w}\) be vectors in \(\mathbb{R^3}\) and \(k\in\mathbb{R}\). (a) \((k\mathbf{u})\times\mathbf{v}=\) k \ ((\mathbf{u}\times\mathbf{v})=\mathbf{u}\times(k\mathbf{v})\). \ (\hspace{2cm}\) (What happens to the determinant above if one row is multiplied by the scalar \(k\)?) Correct answer, well done. Marks for this submission: 1.00/1.00. (b) \(\mathbf{u}\times(\mathbf{v}+\mathbf{w})= (\mathbf{u}\times\mathbf{v})+(\mathbf{u}\times\mathbf{w})\). (c) \((\mathbf{u}+\mathbf{v})\times\mathbf{w}= (\mathbf{u}\times\mathbf{w})+(\mathbf{v}\times\mathbf{w})\). (d) \(\mathbf{u}\times\mathbf{0}=\) 0 0 0 \ (=\mathbf{0}\times\mathbf{u}\). Correct answer, well done. Marks for this submission: 1.00/1.00. (e) \(\mathbf{u}\parallel\mathbf{v} \,\, \Leftrightarrow \,\, \mathbf{u}\times\mathbf{v}=\) 0 0 0 . (Recall: \(\mathbf{u}\parallel\mathbf{v} \) means that \(\mathbf{u}\) and \(\mathbf{v}\) are parallel.) (What happens to the determinant above if \(\mathbf{u}\) and \(\mathbf{v}\) are parallel?) Correct answer, well done. Marks for this submission: 1.00/1.00. (f) \(\mathbf{u}\cdot(\mathbf{u}\times\mathbf{v})=\) 0 and \ (\mathbf{v}\cdot(\mathbf{u}\times\mathbf{v})=\) 0 . Thus, \(\mathbf{u}\times\mathbf{v}\) is to both \ (\mathbf{u}\) and \(\mathbf{v}\). (What are the determinants of the associated 3x3 matrices for these scalar triple products?) Correct answer, well done. Marks for this submission: 1.00/1.00. Correct answer, well done. Marks for this submission: 1.00/1.00. Correct answer, well done. Marks for this submission: 1.00/1.00. (g) For \(\mathbf{u}\nparallel\mathbf{v}\) the ordered triple of vectors \((\mathbf{u},\mathbf{v},\mathbf{u}\times\mathbf{v})\) has right hand orientation . i.e. if using your right hand, you point your index finger in the direction of \(\mathbf{u}\) and your middle finger in the direction of \(\mathbf{v}\) then \(\mathbf{u}\times\mathbf{v}\) will point in the same direction as your thumb if you point it so that it is perpendicular to the plane made by your index and middle fingers. orthogonal

3/27/24, 1:25 AM Determinants: Properties and Applications: Attempt review | eClass https://eclass.srv.ualberta.ca/mod/quiz/review.php?attempt=15149458&cmid=7601901 19/22 Question 13 Correct Mark 9.00 out of 9.00 Example: Let \(\mathbf{u}\) and \(\mathbf{v}\) be vectors in \(\mathbb{R}^3\) and suppose \(\mathbf{u} \times \mathbf{v} = [2,1,-3]\). Compute \((4\mathbf{v} + \mathbf{u}) \times (2\mathbf{u})\). \((4\mathbf{v} + \mathbf{u}) \times (2\mathbf{u})\) \(= ((4\mathbf{v}) \times (2\mathbf{u})) + (\mathbf{u} \times (\mathbf{2u}))\) \(\big(\)by Property \(\big)\) Correct answer, well done. Marks for this submission: 1.00/1.00. \(=\) 8 \((\mathbf{v} \times \mathbf{u}) + 2 (\mathbf{u} \times \mathbf{u})\) \(\big(\)by Property \(\big)\) Correct answer, well done. Marks for this submission: 1.00/1.00. Correct answer, well done. Marks for this submission: 1.00/1.00. \(=\) -8 \((\mathbf{u} \times \mathbf{v}) + 2 (\mathbf{u}\times \mathbf{u}) \) \(\big(\)by Property \(\big)\) Correct answer, well done. Marks for this submission: 1.00/1.00. Correct answer, well done. Marks for this submission: 1.00/1.00. \(=\) -8 \((\mathbf{u} \times \mathbf{v}) + 2\) 0 0 0 \(\big(\)by Property \(\big)\) Correct answer, well done. Marks for this submission: 1.00/1.00. Correct answer, well done. Marks for this submission: 1.00/1.00. Correct answer, well done. Marks for this submission: 1.00/1.00. \(=\) -16 -8 24 Correct answer, well done. Marks for this submission: 1.00/1.00. (c) (a) (h) (e)

3/27/24, 1:25 AM Determinants: Properties and Applications: Attempt review | eClass https://eclass.srv.ualberta.ca/mod/quiz/review.php?attempt=15149458&cmid=7601901 20/22 Question 14 Correct Mark 5.00 out of 5.00 A correct answer is: "(c)" A correct answer is \( 8 \), which can be typed in as follows: 8 A correct answer is: "(a)" A correct answer is \( -8 \), which can be typed in as follows: -8 A correct answer is: "(h)" A correct answer is \( -8 \), which can be typed in as follows: -8 A correct answer is \( \left[\begin{array}{ccc} 0 & 0 & 0 \end{array}\right] \). A correct answer is: "(e)" A correct answer is \( \left[\begin{array}{ccc} -16 & -8 & 24 \end{array}\right] \). Exercise: Let \(\mathbf{u}=[3,-2,1]\), \(\mathbf{v}=[1,1,1]\) and \(\mathbf{w}=[2,-2,0]\). The area of the parallelogram formed by \(\mathbf{u}\) and \(\mathbf{v}\) is \(\mathcal{A}=\sqrt{\,}\) 38  . The volume of the parallelepiped formed by \(\mathbf{u},\,\mathbf{v}\) and \(\mathbf{w}\) is \(\mathcal{V}=\) 2  . Given that the volume of a parallelepiped is the area of its base times its height, the height of the parallelepiped formed by \ (\mathbf{u},\,\mathbf{v}\) and \(\mathbf{w}\) when its base is viewed as the parallelogram formed by \(\mathbf{u}\) and \(\mathbf{v}\) is \(h=\) 2  \(/\sqrt{\,}\) 38  . The parallelepiped formed by \(\mathbf{u}\) and \(\mathbf{v}\) and \(k\mathbf{u}\times\mathbf{v}\) will have the same volume as the parallelepiped formed by \(\mathbf{u},\,\mathbf{v}\) and \(\mathbf{w}\) if \(k=\pm1/\) 19  . Correct Marks for this submission: 5.00/5.00.