Topic 6: ALGEBRA

1. Topic 6: ALGEBRA Matrices and Matrix Operations Determinants and Inverse Matrices System of linear equations Rouché-Fröbenius Theorem Cramer’s Rule Gauss’s Rule

2. MATRICES AND MATRIX OPERATIONS • We will call Matrix of order mxn over the set of the Real numbers (R,+,.) a set of mxn elements of R ordered in m rows and n columns: • We say that a matrix is square if m=n. In this case, we will call traza   • Tr(A)= a11+a22+...+ann • We say that a matrix is diagonal if aij=0, if ij. A matrix is null if aij=0, i,j  • A row is a matrix 1xn and a column is a matrix mx1. We say that a matrix is symmetric if aij=aji

3. We will denote the set of the matrices mxn by Mmxn. Given a Matrix A we call its transpose • If A, B Mmxn, it is possible to calculate A+B: Given A= (aij), B=(bij), the A+B=(cij), where cij= aij+bij, i=1,2,...,m; j=1,2,...,n • If R .A=(cij), where cij= aij , i,j • Given a matrix A Mmxn and B Mnxp, the product of A and B is denoted by A.B=(cij)mxp, where

4. Properties: 1. A.(B.C) = (A.B).C 2. A.(B+C)=A.B+A.C 3. A.BB.A4. If Anxn is a square matrix, I.A=A, where I is the diagonal matrix, with aii=1, i=1,2,..., n and aij=0, i  j Definition:The inverse matrix A-1 of a square matrix A is a matrix such that A-1.A= IDefinition:Given a square matrix Anxn, we call its Determinant, and we will denote as |A|,where by {p1,...,pn} we denote the possible permutations of the number 1,2,... n and by [p1,...,pn] the number of translations used to arrive from 1,2,…,n to this permutation.

5. In particular • Sarrus’s Rule: • We will call the adjunct of the element aij , and denote as • ad(aij)=(-1)i+j.|Aij|, • where |Aij| is the determinant of order (n-1)x(n-1) matrix obtained by canceling the row i and the column j. • The adjunct matrix of A is • The matrix inverse of A is obtained by the formula

6. Properties: (A.B)-1= B-1.A-1 • (A-1)-1=A • (AT)-1=(A-1)t •  Given a matrix A we call minors of order k (km,n) of A the determinants of the sub-matrices constructed with elements belonging to k rows and k columns of A. • We call Range of A, rg(A) the order of the biggest minor of A different from zero.

7. Determinant Properties • |A.B| = |A|.|B|; kN, k0, |Ak|=|A|k ; |A-1|= 1/|A| • |aA! = an|A| • If A has a row (or a column) with all numbers equal to 0, |A|=0 • If A has two rows (or two columns) proportional or equal, then |A|=0 • If in a matrix A, two rows (or two columns) are exchanged then the determinant of the obtained matrix is equal to -|A| • If in a matrix A, to one row (or column) we add a linear combination of other parallel rows (or columns), the determinant of the obtained matrix is |A|

8. SYSTEMS of LINEAR EQUATIONS • We will call a System of Linear Equation with n unknowns a system of m equations as • This system can be written as or A.X=B where A is the coefficients matrix, X is the variable matrix, and B is the independent terms matrix.

9. Classification: • Determined Compatible System: One solution • Undetermined Compatible System: More than one solution • Incompatible System: No solutions. • If we call M= (A|B) the amplified matrix, • THEOREM (ROUCHÉ-FRÖBENIUS) • The system AX=B has solution iff rg(A) = rg(M) • If rg(A)rg(M) then the system is Incompatible.

10. CASES: 1. rg(A)=rg(M) = m= n. Compatible system. The solution is unique and we can apply the Cramer’s Rule: where the determinant in the numerator is obtained by substituting the column i of A by the column of the independent terms. 2.     rg(A)=rg(M)=n<m. It is possible to cancel the equations which are linearly dependent of the others, and then we work as in the case 1. 3. rg(A)=rg(M)=r<n.The terms corresponding to variables that we have not used in the study of the range pass to the second term of the equations. Then, we will proceed as in cases 1 or 2.

11. Solve the system Sol (1,0,-1,2)