Question 1 [10 points] Determine the values of a for which the following system of linear equations has no solutions, a unique solution, or infinitely many solutions. You can select 'always', 'never', 'a = ', or 'a #, then specify a value or comma-separated list of values. xitax2-3x3 = -1 -X1 X2+x3 = -5 -2x1-3x2+3x3 = -6 No Solutions: Always Unique Solution: Always Infinitely Many Solutions: AlwaysQuestion 2 [4 points] Given the determinants of the following matrices: 2 26 -2 14 2 6 -2 2 2 6 14 2 2 14 -2 2 14 6 -2 3 4 10 -1 14 4 10 -1 3 4 10 14 det =2, det =2 , det 3 4 14 -1 =-10, det =-2, det 3 14 10 -1 =8 -2 1 -2 10 -46 1 -2 10 -2 1 -2 -46 -2 1 -46 10 -2 -46 -2 10 1 3 6 6 -23 3 6 6 1 3 6 -23 1 3 -23 6 1 -23 6 6 Find the (unique) solution to the following system: 2x7+2x2+6x3-2x4 = 14 3x7+4x2+10x3-x4 = 14 -2xy+x2-2x3+10x4 = -46 x1+3x2+6x3+6x4 = -23 X=Question 3 ['10 points] Compute the following matrix inverse: You can resize a matrix (when appropriate] by clicking and dragging the bottom-right corner of the matrix. 0 1 2 '1 0 0 0 001 =000 113 0003 Question 4 [10 points] Express the following invertible matrix A as a product of elementary matrices: You can resize a matrix (when appropriate) by clicking and dragging the bottom-right corner of the matrix. 3 A = Number of Matrices: 1 0 0 0 A = 0 0 0Question 5 [10 points] Let L3 be the line passing through the points [21:62: 3_. 3) and Qg=(2= 5= 9) and let L3 be the line passing through the point PE: {16: 14: 30) with direction vector E=[2= 2, 3]: Determine whether L1 and L2 intersect. If so: nd the point of intersection Q. Question 6 [10 points] Let L3 be the line passing through the points (23:65.. '2; =1) and Q2=(7; [1 6). Find a value ofk so the line L2 passing through the point P:- =P;.(19._ 3; k) with direction vector 3+=[6.. 3; 10]T intersects with L3. k = U Question 7 ['10 points] Let L be the line passing through the point P=(1__ -Ll_. l) with direction vector H'=[{]__ I; 3]T. (4: 3, 3] to L. and the point Q on L that is closest to P0. Use the square root symbol "-.-" whe (i=0 Q=(U..0.0} :in the shortest distance d from the point P0: e needed to give an exact value for your answer. Question 8 [10 points] Let L be the line through the point Q = (0, -7, 5) and direction vector d=[-1, -2, 2] I Find the two distinct points R1 and R2 on L at distance 6 \\2 to the point P = (-1, -3, -5). R1 = (0, 0, 0) R2 = (0, 0, 0)Question 9 [10 points] Let L be the line passing through the point P=(-4, 1, 3) with direction vector d=[4, -1, 0], and let T be the plane defined by -4x-5y+5z = 42. Find the point Q where L and T intersect. Q = (0 , 0, 0)Question 10 [10 points] Let S be the plane defined by xy = 6. and let T be the plane defined by x5.z =2. Find the vector equation for the line where S and T intersect. O U D+tU D 0 \"ME: >4. || Question 11 [10 points] Consider the following matrix A and its reduced row-echelon form: 6 -18 -18 30 -12 -12 1 -3 -3502 2 -6 -6 10 -4 -4 0 0 0 012 A = rref(A) = 2 -6 10 -2 0 0 0 0 000 -6 18 18 -30 3 -6 0 0 0 000 Find the dimensions of row(A), null(A), and col(A), and give a basis for each of them. Dimension of row(A): 1 O Basis for row(A): Dimension of null(A): 1 Basis for null(A): Dimension of col(A): 1 Basis for col(A):Question 12 [10 points] Consider the following matrix: 3 6 -3 -9 6 A= -3 -6 3 9 -1 3 6 -2 -10 5 Give a basis for each of col(A) and null(A). Number of Vectors: 1 Basis for col(A) 0 O Number of Vectors: 1 Basis for null(A)Question 13 [10 points] Find a matrix A that induces the transformation T:R-R- given below. X x-6y-z = x-2y-7z Z 0 0 0 A = 00 0 0 0 0Question 14 [10 points] If TEE3 is a linear transformation and the action of T on the special vectors [I and 1L7 is as given: find a formula for RX}: where X is any . _.\".I vector In 3". 3 2 4 3 11: V: mo: 3 TWJ= 6 3 2 6 4 0 l" r = 0 y Question 15 [10 points] Suppose T:R*-R* is the transformation given below. Determine whether T is one-to-one and/or onto. If it is not one-to-one, show this by providing two vectors that have the same image under T. If T is not onto, show this by providing a vector in R* that is not in the range of T. XO 6x2-6X3 X1 2x2-4x3 T = X2 xo-3x1+x2-3x3 X3 7x2-2x3 T is one-to-one T is ontoQuestion 16 [10 points] Find an invertible matrix P and a diagonal matrix D such that PAP=D. 12 0 5 A = 0 -20 -30 0 13 0 0 0 0 0 0 P = 0OO D= 00 0 0 0 0 0 0 0Question 1? [10 points] Find an invertible matrix P and a diagonal matrix D such that P_1_4P=D. 14 02 _2? 02 _2922 41404 0000 0000 _0000 _0000 _0000 _0000 0000 0000 Question 13 [10 points] Find a formula in terms of k for the entries of Elk. where A is the diagonalizable matrix below and P_1A=1P=Dfor the matrices P and D below. 13 10 3 2 3 0 A: P: D: [15 42] [a a] 0 a] 00 00 e Question 19 [10 points] Find a formula in terms of k for the entries of A*, where A is the diagonalizable matrix below, and the eigenvalues of A are -3 and 3. -9 6 A = -12 9 0 0 AK = 0 0