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00
1 2
00 in row echelon form? 0
1
0
0 1
3 1
0 F= 04
0 2
11 011
100 1203
G = 0 0 1 0 0000 Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Gaussian Elimination The elementary row operations that can be performed on a
matrix A are:
1
2 3 Interchange two rows of A.
Replace a row r of A with λr , where λ is a nonzero real
number.
Replace a row r of A with r + λr , where λ ∈ R and r is
another row of A. Gaussian elimination is the process of using elementary row
operations to change a matrix A into a matrix R , with R in
row echelon form.
One can solve a system of linear equations using Gaussian
elimination together with back substitution. Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Examples
Example
Use Gaussian elimination and back substitution to solve the linear
systems.
1 3x1 + 2x2 − x3 = 4
x1 − 2x2 + 2x3 = 1
11x1 + 2x2 + 4x3 = 14
2 x1 + x2 = 1
x1 − x2 = 3
−x1 + 2x2 = −2
(This system is overdetermined) Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Reduced Row Echelon Form of a Matrix A matrix A is said to be in reduced row-echelon form if A has
the following properties.
1
2 A is in row echelon form.
If row i is nonzero with leading 1 in column j , then column j
has only one nonzero entry, namely the leading 1 in row i . Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Examples
Which of 1
0
A=
0 2
0
B=
0 0
C = 0
0 the following matrices are 23
1 2 12
01
0 0
D=
0 0 03
00
1 2
00 10
0 1
E= 13
00
1 2
00 in reduced row echelon form? 0
1
0
0 1
3 1
0 F= 04
0 2
11 011
100 1203
G = 0 0 1 0 0000 Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Gauss-Jordan Elimination Gauss-Jordan elimination is the process of using elementary
row operations to change a matrix A into a matrix R , with R
in reduced row echelon form.
The reduced row ech...

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