MATH-225 Final Exam — 20.05.2021 — 13:00–16:00
N.B. Correct answers without sufficient correct mathematical explanations will not get full credit.
Q 1: Let A be an n × n matrix, λ1 an eigenvalue of A, and let In denote the identity matrix of size n × n. Recall
that the multiplicity of λ1 is the largest integer k such that (λ − λ1 )k is a factor of the characteristic polynomial
|λIn − A|.
(a) (5 pts) Show by an example that the dimension of Null(λ1 In − A) can be different from the multiplicity of
λ1 .
(b) (5 pts) If A and B are two similar square matrices, show that they have the same characteristic polynomial.
(c) (5 pts) Let A be an 8 × 8 matrix, let {v1 , v2 , . . . , v8 } be a basis of R8 such that {v1 , v2 , v3 , v4 } is a basis of
Null(I8 − A), and let S be the 8 × 8 matrix with columns v1 , v2 , . . . , v8 in this order. Calculate the leftmost 4
columns of the matrix S −1 AS.
(d) (5 pts) What is the minimal multiplicity of the eigenvalue 1 that the matrix A in (c) may have?
Q 2: (a) (8 pnts) Show that the equation y 2 dx + (2xy − y 2 )dy = 0 is exact and then find the general solution.
We have learnt four different methods to solve first order differential equations of the types: separable, linear,
Bernoulli, and homogeneous.
dy
(b) (6 pnts) and (c) (6 pnts): Solve the initial value problem (2x − y) dx + y = 0, y(1) = 1, by using any two
of these different methods.
4 −5 1
Q 3: (a) (12 pnts) Find all eigenvalues and associated eigenvectors of A = 1 0 −1.
0 1 −1
dx
(b) (4 pnts) Find 3 linearly independent solutions to the system of differential equations = Ax, where
T dt
x(t) = x1 (t) x2 (t) x3 (t) .
(c) (4 pnts) Check that your solutions in part (b) are actually linearly independent using the Wronskian.
Q 4: (a) (15 pts) Show that y1 = x is a solution to the differential equation (1 + x2 )y 00 − 2xy 0 + 2y = 0, for
x ∈ (0, 1). Find a basis for the solution space of this ODE.
(b) (5 pts) The roots of the characteristic equation of a homogeneous linear ODE of order 5 are as follows:
(i) 0 with multiplicity 3, (ii) 1 with multiplicity 1, (iii) 3 with multiplicity 1. Find a general solution to the
corresponding differential equation.
Q 5: (20 pts) Solve the following nonhomogeneous differential equation y 00 − y = ex · sin(x).
N.B. Correct answers without sufficient correct mathematical explanations will not get full credit.
Q 1: Let A be an n × n matrix, λ1 an eigenvalue of A, and let In denote the identity matrix of size n × n. Recall
that the multiplicity of λ1 is the largest integer k such that (λ − λ1 )k is a factor of the characteristic polynomial
|λIn − A|.
(a) (5 pts) Show by an example that the dimension of Null(λ1 In − A) can be different from the multiplicity of
λ1 .
(b) (5 pts) If A and B are two similar square matrices, show that they have the same characteristic polynomial.
(c) (5 pts) Let A be an 8 × 8 matrix, let {v1 , v2 , . . . , v8 } be a basis of R8 such that {v1 , v2 , v3 , v4 } is a basis of
Null(I8 − A), and let S be the 8 × 8 matrix with columns v1 , v2 , . . . , v8 in this order. Calculate the leftmost 4
columns of the matrix S −1 AS.
(d) (5 pts) What is the minimal multiplicity of the eigenvalue 1 that the matrix A in (c) may have?
Q 2: (a) (8 pnts) Show that the equation y 2 dx + (2xy − y 2 )dy = 0 is exact and then find the general solution.
We have learnt four different methods to solve first order differential equations of the types: separable, linear,
Bernoulli, and homogeneous.
dy
(b) (6 pnts) and (c) (6 pnts): Solve the initial value problem (2x − y) dx + y = 0, y(1) = 1, by using any two
of these different methods.
4 −5 1
Q 3: (a) (12 pnts) Find all eigenvalues and associated eigenvectors of A = 1 0 −1.
0 1 −1
dx
(b) (4 pnts) Find 3 linearly independent solutions to the system of differential equations = Ax, where
T dt
x(t) = x1 (t) x2 (t) x3 (t) .
(c) (4 pnts) Check that your solutions in part (b) are actually linearly independent using the Wronskian.
Q 4: (a) (15 pts) Show that y1 = x is a solution to the differential equation (1 + x2 )y 00 − 2xy 0 + 2y = 0, for
x ∈ (0, 1). Find a basis for the solution space of this ODE.
(b) (5 pts) The roots of the characteristic equation of a homogeneous linear ODE of order 5 are as follows:
(i) 0 with multiplicity 3, (ii) 1 with multiplicity 1, (iii) 3 with multiplicity 1. Find a general solution to the
corresponding differential equation.
Q 5: (20 pts) Solve the following nonhomogeneous differential equation y 00 − y = ex · sin(x).
