ft-uiowa-math2550 Assignment HW8fall14 due 10/23/2014 at 11:59pm CDT 3. (1 pt) local/library/ui/fall14/hw8 3.pg Given the matrix

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1 me me Assignment HW8fall4 due /23/24 at :59pm CDT ft-uiowa-math Calculate the determinant of D - E F 2 I 4 J 5 C 2 ( pt) local/library/ui/fall4/hw8 2pg Evaluate the following 3 3 determinant Use the properties of determinants to your advantage -4-3 D - E F 2 I 4 J none of the above ( pt) local/library/ui/fall4/hw8 3pg Given the matrix (a) find its determinant 25-5 C -4 D -2 E - F I 5 J 7 (b) does the matrix have an inverse? No Yes 4 ( pt) local/library/ui/fall4/hw8 4pg If A and B are 4 4 matrices det(a) = 4 det(b) = 3 then det(ab) = Does the matrix have an inverse? No Yes E -5-2 C - D -8 E -5 F 3 H 6 I 8 J 2

2 det(2a) = det(a T ) = D -2 E - F - H 2 I 28 J 36 K 4 L None of those above -3-2 C - D E F 2 3 H 4 det(b ) = det(b 4 ) = -5-4 C D E F 4 5 H H E 5 ( pt) local/library/ui/fall4/hw8 5pg Find the determinant of the matrix - A = det(a) = D -2 E F 2 4 H 24 I 36 J 4 6 ( pt) local/library/ui/problem7pg A and B are n n matrices Adding a multiple of one row to another does not affect the determinant of a matrix True False D E 2 F 36 8 H 24 2 If the columns of A are linearly dependent then deta = True False

3 det(a + B) = deta + detb True False 7 ( pt) local/library/ui/fall4/hw8 7pg Suppose that a 4 4 matrix A with rows v v 2 v 3 and v 4 has determinant deta = 4 Find the following determinants: B = C = D = v v 2 v 3 2v 4 det(b) = -8-5 D -8 E -9 F 9 H 2 I 5 J 8 v 4 v v 2 v 3 det(d) = det(c) = -8 4 C -9 D -3 E F 3 9 H 2 I 8 J None of those above v v 2 v 3 + 6v 4 v C -9 D -3 E F 3 9 H 2 I 8 J None of those above D 8 ( pt) local/library/ui/fall4/hw8 8pg Use determinants to determine whether each of the following sets of vectors is linearly dependent or independent Linearly Dependent Linearly Independent Linearly Dependent Linearly Independent -3-9 Linearly Dependent Linearly Independent Linearly Dependent Linearly Independent

4 9 ( pt) local/library/ui/fall4/hw8 pg A = The determinant of the matrix is C -63 D 25 E -2 F 324 H 63 I 24 J None of those above ( pt) local/library/ui/fall4/hw8 2pg A = And now for the grand finale: The determinant of the matrix is C D 5 E 2 F I None of the above Hint: Find a good row or column and expand by minors D ( pt) local/library/ui/fall4/hw8 pg Findthe determinant of the matrix 2 2 M = det(m) = D -5 E 5 F 8 2 H 8 C Hint: Remember that a square linear system has a unique solution if the determinant of the coefficient matrix is non-zero Solution: Since all the rows are the same a linear system with A as its coefficient matrix cannot have a unique solution and therefore the determinant of A is zero C 2 ( pt) local/library/ui/43apg Find bases for the column space and the null space of matrix A You should verify that the Rank-Nullity Theorem holds An equivalent echelon form of matrix A is given to make your work easier A = Basis for the column space of A = Basis for the null space of A = 4

5 SOLUTION: A basis for the column space determined from the pivot columns and 2 is Solve Ax = to obtain x = s 3 3 and so the nullspace basis is 3 3 \mbox{} \cr \mbox{3} \cr \mbox{2} \cr \(\displaystyle\left\begin{array}{c} \mbox{2} \cr \mbox{3} \cr \mbox{6} \cr \mbox{-3} \cr \mbox{-3} \cr \mbox{} \cr 3 ( pt) local/library/ui/433pg Find bases for the column space and the null space of matrix A You should verify that the Rank-Nullity Theorem holds An equivalent echelon form of matrix A is given to make your work easier A = Basis for the column space of A = Basis for the null space of A = SOLUTION: A basis for the column space determined from the pivot columns and 2 is 2 5 Solve Ax = to obtain x = s the nullspace basis is s 2 + and so \mbox{} \cr \mbox{} \cr \(\displaystyle\left\begin{array}{c} \mbox{} \cr \mbox{} \cr \mbox{} \cr \mbox{4} \cr \mbox{-5} \cr \mbox{} \cr \mbox{} \cr \(\displaystyle\left\begin{array}{c} \mbox{3} \cr \mbox{} \cr \mbox{} \cr \mbox{} \cr 4 ( pt) Library/Rochester/setLinearAlgebraBases/ur la 3pg Find a basis of the column space of the matrix A = \mbox{} \cr \mbox{} \cr \(\displaystyle\left\begin{array}{c} \mbox{} \cr 5 ( pt) Library/WHFreeman/Holt linear algebra/chaps -4- /427pg Find the null space for A = What is null(a)?

6 { } span R 3 C span 7 6 D R 2 E span { } F span { } H none of the above SOLUTION A is row reduces to The basis of the null space has one element for each column without a leading one in the row reduced matrix Thus Ax = has a zero dimentional null space and null(a) is the zero vector 6 ( pt) local/library/ui/423pg 3 Find the null space for A = 8 What is null(a)? { } 8 span span 8 C R 2 D R 3 E span 8 F span 8 { } span 8 H none of the above 6 SOLUTION A is row reduced The basis of the null space has one element for each column without a leading one in the row reduced matrix Thus Ax = has a one dimentional null space and thus null(a) is the subspace generated by 3 8 F 7 ( pt) Library/WHFreeman/Holt linear algebra/chaps -4- /4347pg Indicate whether the following statement is true or false? If A and B are equivalent matrices then col(a) = col( B) SOLUTION: FALSE Consider A = F ( pt) local/library/ui/fall4/hw7 27pg Determine the rank and nullity of the matrix The rank of the matrix is -4-3 D - E F 2 I 4 J none of the above The nullity of the matrix is -4-3 D - E and B =

7 F 2 I 4 J none of the above SOLUTION: When reduced to row-echelon form there are two non-zero rows so the rank of the matrix is 2 and the nullity is Generated by c WeBWorK Mathematical Association of America 7