01 System of Linear Equations
Elementary Row Operations¶
-
Any 2 rows can be interchanged
\(R_1 \iff R_2\) - Any row can be multiplied/divided by any number other than 0
\(R_1 \to 2R_1\) - Any row can be added/subtracted to any row
\(R_1 \to R_1 \pm 2 R_2\)
REF¶
Reduced Echelon Form
Upper \(\triangle\)r matrix
-
1st non-zero elment in a row should be 1
(called as leading one) - Leading one should occur to the right side of previous rows’ leading one(s) - If there is any zero row, it should be the last row otherwise, we need to interchange rows to ensure this rule
example
RREF¶
diagonal matrix
is the REF matrix where the elements of the columns of the leading ones (other than itself) are 0.
Rank¶
no of non-zero rows of a matrix in REF/RREF
Gauss Methods¶
| Method | Form |
|---|---|
| Gauss Elimination | REF |
| Gauss Jordan | RREF |
- Write equation in matrix form \(AX = B\), where
- \(A\) is coefficients matrix
- \(B\) is constant matrix
- \(X\) is variable matrix
Converted augmented matrix = \([A | B]\) into REF
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Cases
\(n\) is the number of unknown variables
| Rank(A\vert B) | |
|---|---|
| \(\ne\) rank(A) | no solutions |
| \(=\) rank(A) \(= n\) | unique solutions |
| \(=\) rank(A) \(< n\) | infinite solutions |
-
Back Substitution
Degree of freedom = no of vars - no of equations
Homogeneous Linear System¶
There will always be a solution.
If there is unique solution, it is always all 0s. This is called as trivial solution.
Inverse of matrix¶
If \(A\) and \(B\) are 2 non-singular matrices such that \(|A| \ne 0\), then \(A^{-1} = B \iff A\cdot B = I\)
\(I\) is identity matrix
To find inverse
- use row transformations to convert \([A:I] \to [I:B]\)
- then \(B = A^{-1}\)
If \(A\) is singular, inverse does not exist