Скачать или смотреть MATRICES(LEC-13)(Ch-2)COND. OF CONSISTENCY AND NON HOMOGENEOUS EQ CO-EFF. MATRIX (SEM-01)

MATRICES(LEC-13)(Ch-2)COND. OF CONSISTENCY AND NON HOMOGENEOUS EQ CO-EFF. MATRIX (SEM-01)#bbmku
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#Unit _2 lec_12
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A system of linear non-homogeneous equations has the form:

Matrix Form

AX = B

where:

1. A is the coefficient matrix (m x n)
2. X is the variable matrix (n x 1)
3. B is the constant matrix (m x 1)

Condition of Consistency

For the system to have a solution, the following condition must be satisfied:

1. *Rank(A) = Rank([A|B])*, where [A|B] is the augmented matrix.

If Rank(A) ≠ Rank([A|B]), the system is inconsistent (no solution).

Nature of General Solution

If the system is consistent:

1. *Unique solution*: If Rank(A) = n (number of variables), the system has a unique solution.
2. *Infinite solutions*: If Rank(A) less than n, the system has infinitely many solutions, represented by X = Xp + Xt, where:
3. Xp is a particular solution
4. Xt is the general solution of the homogeneous system (AX = 0)

Types of Solutions

1. *Trivial solution*: X = 0 (when B = 0)
2. *Particular solution*: Xp (satisfies AX = B)
3. *Homogeneous solution*: Xt (satisfies AX = 0)
4. *General solution*: X = Xp + Xt

Methods to Find Solutions

1. Gaussian Elimination
2. LU Decomposition
3. Matrix Inversion
4. Cramer's Rule (for 2x2 and 3x3 systems
Consistency

1. *Definition*: A system of linear equations AX = B is consistent if it has at least one solution.
2. *Condition*: Rank(A) = Rank([A|B]), where [A|B] is the augmented matrix.
3. *Types of Consistency*:
*Unique solution*: Rank(A) = n (number of variables)
*Infinite solutions*: Rank(A) less than n

Inconsistency

1. *Definition*: A system of linear equations AX = B is inconsistent if it has no solution.
2. *Condition*: Rank(A) ≠ Rank([A|B])
3. *Characteristics*:
No solution exists
System is contradictory

Examples

1. *Consistent System*:

2x + 3y = 7
4x + 6y = 14

Rank(A) = Rank([A|B]) = 1, so the system is consistent.

1. *Inconsistent System*:

2x + 3y = 7
4x + 6y = 16

Rank(A) = 1, Rank([A|B]) = 2, so the system is inconsistent.

Theorems

1. *Rouché-Capelli Theorem*: A system AX = B is consistent if and only if Rank(A) = Rank([A|B]).
2. *Frobenius Theorem*: A system AX = B has a unique solution if and only if |A| ≠ 0.

Methods to Check Consistency

1. Gaussian Elimination
2. LU Decomposition
3. Matrix Inversion
4. Cramer's Rule
5. Rank calculation using row/column operations

Key Formulas

1. Rank(A) = Number of linearly independent rows/columns
2. |A| = Determinant of matrix A
3. [A|B] = Augmented matrix
A system of linear non-homogeneous equations with a coefficient matrix containing λ (lambda) or μ (mu) can be represented as:

AX = B

where:

A = Coefficient matrix with λ or μ
X = Variable matrix
B = Constant matrix

To solve such a system, we can follow these steps:

1. Write the coefficient matrix A with λ or μ.
2. Find the determinant of matrix A, |A|.
3. If |A| ≠ 0, the system has a unique solution.
4. If |A| = 0, the system may have infinitely many solutions or no solution.

Let's consider an example:

Suppose we have a system of linear non-homogeneous equations:

λx + 2y = 3
x + (λ + 1)y = 4

The coefficient matrix A is:

A = | λ 2 |
| 1 λ+1|

To find the determinant of matrix A:

|A| = λ(λ + 1) - 2

Now, let's analyze the system:

If |A| ≠ 0, the system has a unique solution.

If |A| = 0, we need to examine the augmented matrix:

[A|B] = | λ 2 | 3 |
| 1 λ+1| 4 |

If Rank(A) = Rank([A|B]), the system has infinitely many solutions.

If Rank(A) ≠ Rank([A|B]), the system has no solution.