Systems of Linear Equation and Matrices. CHAPTER 1 FASILKOM UI 05. YR. Introduction ~ Matrices. Information in science and mathematics is often organized into rows and columns to form rectangular arrays. Tables of numerical data that arise from physical observations
Systems of Linear Equation and Matrices
FASILKOM UI 05
Introduction to Systems
of Linear Equations
where a1, a2, & b are real constants,
where a1, …, an & b are real constants
x1, …, xn = unknowns.
no intersection, no solution
at only one point: one solution
infinite many points of intersection,
infinitely many solutions
a11x1 + a12x2 + ... + a1nxn = b1
a21x1 + a22x2 + ... + a2nxn = b2
am1x1 + am2x2 + ... + amnxn = bm
Matrices and Matrix Operations
Inverses; Rules of Matrix Arithmetic
Theorem: Properties of
A+B = B+A
A+(B+C) = (A+B)+C
A(BC) = (AB)C
A(B+C) = AB+AC
(B+C)A = BA+CA
A(B-C) = AB-AC
(B-C)A = BA-CA
a(B+C) = aB+aC
a(B-C) = aB-aC
(Commutative law for addition)
(Associative law for addition)
(Associative for multiplication)
(Left distributive law)
(Right distributive law)
(a+b)C = aC+bC
(a-b)C = aC-bC
a(bC) = (ab)C
a(BC) = (aB)C
is invertible if ad-bc ≠ 0, in which case the inverse is given by the formula
A0=IAn = AA...A(n>0)
is any polynomial, then we define
Elementary Matrices and a Method for Finding A-1
Assume A is invertible and let x0 be any solution of Ax=0.
Let Ax=0 be the matrix form of the system
Assumed that the reduced row-echelon form of A is In by a finite sequence of elementary row operations, such that:
By theorem, E1,…,En are invertible. Multiplying both sides of equation on the left we obtain:
This equation expresses A as a product of elementary matrices.
If A is a product of elementary matrices, then the matrix A is a product of invertible matrices, and hence is invertible.
Added –2 times the first row to the second and –1 times the first row to the third.
Added 2 times the second row to the third.
Multiplied the third row by –1.
Added 3 times the third row to the second and –3 times the third row to the first.
We added –2 times the second row to the first.