## Homework #8 Consider the veracity or falsehood of each of the following statements. For bonus, argue for those that you believe are true while providing a counterexample for those that you believe are false. If the first and third rows of A are equal, then det A 0. If P is a projection, then uCP if and only if Pu u. If P is a projection, and detP 0, then P I . If A has determinant 10, then 1 A has determinant 1 10 . If B is invertible, 1 1 det(A B ) det A (detB) . If P is a projection, and R 2P I , then 2 R I . If P is a projection, and P I , then detP 0 . Short Computations. All of the following do not involve long computations: Suppose 1 2 1 5 1 8 A and 1 9 2 4 3 1 A . Compute 7 13 19 A . Compute 0 8 7 1 0 2 3 4 5 3 0 9 2 0 0 0 3 0 0 0 1 9 3 2 0 det . Use Cramer’s Rule to find 5 x (hint: you do not need your calculator). 1 2 3 4 5 5x 2x 8x x 3x 13 1 3 3x 5x 0 1 3 5 3x 3x 3x 9 1 2 3 5 3x 2x x 2x 7 1 3 x 4x 0 Let A 1 2 3 4 1 3 4 6 2 5 13 15 4 10 15 31 . Given is that det A 61. Do the following: 1 1 2 4 2 3 5 10 3 4 13 15 4 6 15 31 det det2A 1 3 4 6 2 4 6 8 2 5 13 15 4 10 15 31 det 1 3 4 6 2 5 13 15 4 10 15 31 1 2 3 4 det Consider the matrix A 0 1 0 0 0 0 1 0 0 0 0 1 1 2 2 1 . Use row (or column) expansion to compute det(xI A) . The matrix 4 1 1 2 1 1 1 4 1 1 2 1 1 1 4 1 1 2 2 1 1 4 1 1 1 2 1 1 4 1 1 1 2 1 1 4 1 6 P is the projection matrix for the column space of matrix A. This matrix A is also known to be of full rank. Answer the following, giving reasons for your answers. Find a transparent basis and the dimension for the column space of P. Find a basis and the dimension for the column space of A . What size is the matrix A ? Find a transparent basis and the dimension for the null space of P. Find a transparent basis and the dimension for the row space of P. Find a basis and the dimension for the null space of A. For which of the following b can you find a solution to the system Ax b ? This does not mean you should find a solution, only whether one could or not. 10 17 19 14 10 17 19 14 13 10 17 19 14 13 23 1 1 1 1 1 1 . It is known that certain vector u is a solution to the system Ax c . Give all solutions to Ax c . It is also known that 1 2 3 4 5 6 Ax does not have a solution. How would you change the constant vector so that there would be a solution? Extra Problems. Fill in the blank with the best possible expression to complete the sentence truthfully. Only that one will be counted correct. 1. matrix with two equal columns will have zero determinant. 1 2 3 Some Every No 2. If A is invertible, then A commute with its inverse. 1 2 3 must always can will not 3. If A is 6 9 , then the columns of A be linearly independent. While in AT , the columns be linearly independent. 1 2 3 can have to cannot 4. Let A be square, and suppose Ax 0 has a nontrivial solution. Then detA equal 0. 1 2 3 may cannot must 5. Let A and B be 3 3. Then det (AB) equal det(A)det(B) . 1 2 3 could must couldn’t 6. Let A be square and suppose detA 0. Then have an inverse 1 2 3 will not may must always 7. Let A and B be 2 2 . Then det (A B) equal det(A) det(B) . 1 2 3 could must could not 8. exist a 6 6 matrix all of whose entries are whole numbers and its determinant is 2 5 . 1 2 3 There does There does not There might Bonus: Consider the matrix 0 0 1 0 2 0 n 0 . Give its determinant as a function of n.

info@checkyourstudy.com Whatsapp +919911743277