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
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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.