MATH253-10A + ENGG283-10A Test 1
1. (a) Find a Cartesian equation for the plane which contains the three points (1,2,3), (2,1,2) and (1,1,5).
(b) Find the area of the triangle with corners at the three points in part (a).
(c) Find the point where the line (x, y, z) = (1,0,2) +t(2,1,−1) in- tersects the planex+y+z = 9.
(d) Find the acute angle between the line and the plane in part (c).
2. (a) Solve the following system of equations
y +z = 2
x −y +2z −w = 1 2x +3y −z +3w = 7 (b) Find the adjugate of the matrix
A=
1 1 1 0 1 2 5 2 1
(c) Explain how the adjugate is related to the inverse and demonstrate this relationship using the matrix A above.
1
3. (a) Find the determinant of the matrixM by row reducing it to upper triangular form.
M =
0 1 1 0 1 1 1 1 2 0 1 2 0 2 2 3
(b) Explain what the determinant of a matrix tells us about the nature of the matrix.
(c) Express the matrixC as a product of elementary matrices where C =
0 3 2 4
4. (a) For what values of the constant λ will the following homogeneous system of equations (in matrix form) have non-trivial solutions.
Note that you are not required to solve the system.
(4−λ) 0 1
2 (3−λ) 2
1 0 (4−λ)
x y z
=
0 0 0
(b) Demonstrate that the following matrix defines a rotation. Find the equation of the axis and determine the angle of rotation.
R = 1 5
4 3 0
0 0 5
3 −4 0
(c) Explain why the following set of vectors is NOT a basis of R3
1 1 1
,
1 2 3
,
2 3 4
2
