Mathematics (Part 2) - A.Y. 2013/2014
Silvia Faggian
Suggestion for self studying – 2.5
(updated 27.11.2013)
Main content
The study material here examined refers to the 4th and 5th week of the course (lectures and PS held on
29 November and 3, 4, 12 December 2012). The main contents are: matrix operations, determinants,
inverse matrices, linear systems.
Important notions
Topic
[Section] Page
Matrix Operations
[15.2-15.4] 548-561
Transpose, symmetric matrices
[15.5] 562-564
Determinants (skip: all of geometric
[16.1-16.5] 585-603
interpretation, and Sarrus rule)
Inverse of a Matrix (skip p.611)
[16.6, 16.7] 604-610
Vectors, inner product
[15.7] 570-573
Rank of a matrix
[lecture notes]
Linear systems; Rouche-Capelli’s Theorem
[15.1, lecture notes]
Cramer’s rule
[16.8] 613-616
Cramer’s rule (adapted)
[lecture notes]
Required abilities
Ability
Warm-up Exercises
Compute the sum of two matrices, the product
[15.2-15.4] #2-4 p.551; #1-3, 5-7 p.555,
by a scalar, the multiplication of matrices
#1, 4-7, pp. 561-562
Compute determinants of any order
[16.1-16.5] #1, 4, 5, 7, pp.587-588
#2, 4-7 pp. 591-592, #1, 5, p.595
#1-12 p.599-600; #1, 2, p.603
Compute the inverse of a matrix
[16.6] #1,2, 4-6, 8, 10, 11, p.609
[16.7] #1-3, p.612
Compute the rank of a matrix
[that of any matrix in the TB]
Solve systems by means of Cramer’s rule
#3, p.588, #3, p.592; #1-3, pp.616
Suggestions for self study
(Important!!)
1. Read and learn the definition of minor of order k of a matrix (“a minor is the determinant of a
square k × k sub-matrix of A”).
2. Definition: the rank of a (rectangular m × n) matrix A is the greatest order of any non-zero
minor in the matrix.
3. Write an procedure for computing the rank of a matrix.
4. Rouché-Capelli’s Theorem: A system of m linear equations with n variables has a solution
if and only if the rank of its incomplete matrix (also said, coefficient matrix) A is equal to the
rank of its complete matrix (also said, augmented matrix) [A|b]. If there are solutions, they form
set of dimension n − r, where r = rank(A) = rank(A|b). In particular:
• if n = r, the solution is unique,
• otherwise there are infinite number of solutions, say ∞n−r , meaning that there is a choice
(among all n variables) of n − r variables that may vary freely in R. (Such variables are
said free variables.)
5. Write an procedure for computing the solutions of a system (of linear equations with n variables)
by means of Cramer’s rule when the common rank r of complete/incomplete matrices is stricly
less than n (r < n).
Theoretical questions
(Questions you can expect at the exam)
1. Write the following definitions:
• transpose of a general m × n real matrix;
• square matrix, of order n;
• symmetric matrix.
For all definitions give an example and a counterexample.
2. Consider matrix multiplication:
• write what dimensional rules two matrices must obey in order to be multiplied;
• cite a property for number multiplication which is false in the case of matrices; exemplify;
• write what it means, for two matrices, to commute; build an example.
3. Define the power of a matrix by recursion.
4. If A and B are two square matrices, and det A and det B are known, then det(AB) = .... ....?
Moreover from the rule, derive:
• that det(A−1 ) = (det A)−1 for any invertible matrix;
• that any invertible matrix A need have det A 6= 0.
5. Define the rank of a general m × n real matrix; add some remarks of your choice. Exemplify.
6. State Rouché- Capelli’s Theorem. Apply it to three examples of your invention, one with no
solutions, one with infinite solutions, one with one solution.