0:00 - 1:30 - Using Matrices to Solve Non-Homogeneous Linear Systems
Matrix Representation
The formula for the system of equations is axe = b.
Matrix Inverse Solution Method
Premultiply on both sides by matrix a's inverse (a⁻¹). If a⁻¹ exists, the solution is x = a⁻¹ * b.
Prerequisite for Inverse
The prerequisite for a⁻¹ existence is: A (det(a)) must have a non-zero determinant.
Alternative Formula
Another way to express the solution is: x = (adjugate of a / determinant of a) * b.
1:30–3:00 - Matrix Method Special Cases (det(a) = 0)
If det(a) = 0
Examine the numerator, which is the adjugate of a * b.
- Inconsistent System (No Solutions): The system is inconsistent if (adjugate of a * b) is not zero.
- Infinite Solutions Consistent System: The system is consistent and has an infinite number of solutions if (adjugate of a * b) is likewise zero.
3:00–5:30 - Using Determinants (Cramer's Rule) to Solve Non-Homogeneous Linear Systems
Method
Cramer's rule, which uses determinants, is an additional approach to solving axe = b.
Formula
The solution for each unknown variable xáµ¢ is given by: xáµ¢ = determinant(Aáµ¢) / determinant(a).
Determinant A
This is the determinant of matrix a, where the elements of matrix b on the right have taken the place of the iáµ—Ê° column. This is a substitution.
Notation
In shorthand, this is also written as Aáµ¢ / a.
Connection to the Matrix Method
It is possible to demonstrate that Cramer's rule and the matrix inverse method are equivalent; they can be shown to work out symbolically.
5:30 - 6:30 - Cramer's Rule Special Cases (det(a) = 0)
If det(a) = 0
Examine the numerator (determinant(Aáµ¢)) once more.
- System Inconsistency (No Solutions): The system is inconsistent if any of the determinants in the numerator (any Aáµ¢) are not zero.
- Infinite Solutions Consistent System: The system is consistent and has an infinite number of solutions if all determinants Aáµ¢ are zero for all i.
6:30 - 7:30 - Homogeneous System
Definition
A homogeneous system is one in which all of the elements on the right side (matrix b) are zero.
- Trivial Solution: There is only a trivial solution if det(a) is non-zero, meaning that all of the variables (x₁, x₂,..., xn) are zero.
- Infinite Number of Solutions: An infinite number of solutions exists if det(a) is zero.
Worked Example: Setting up the Problem
Example Equations
The equations are: x + 7y - 3z = 11, 25y + z = -3, 3x - 6y + 2z = 0.
- First Verifications: There are three unknowns (x, y, z) and three equations. The number of unknowns is equal to the number of equations.
- Due to the presence of non-zero elements (11, -3, 0) on the right-hand side, the system is non-homogeneous.
Matrix Form (axe = b)
Determine which matrices are a, x, and b.
Matrix Definitions
- a = [[1, 7, -3], [3, -6, 2]].
- x = [[x], [y], [z]].
- b = [, [-3], ].
Worked Example: Verifying the 'a' Determinant
Next Step: Verify whether matrix a's determinant is zero.
Calculation Method: To make calculations easier, expand the determinant by a column or row with zero elements (in this case, the first column).
Determinant Calculation
det(a) = 1*(50 - (-6)) - 0*(cofactor) + 3*(7 - (-75)) = 1*56 - 0 + 3*82 = 56 + 246 = 302.
The determinant of a is 302, which is not zero. This indicates that the unknown values (x, y, and z) can be located.
Worked Example: Applying the Matrix Method to Solve
The objective: Use X Y Z = a⁻¹ * b to find X, Y, and Z.
Steps
- To calculate a⁻¹, use the formula a⁻¹ = adjugate(a) / det(a).
Compute Adjugate(a)
Finding each element's cofactor and transposing the matrix of cofactors are necessary for this. One exercise that is mentioned is the computation of cofactors (e.g., c₁₁ = 56).
Adjugate(a) Result
The adjugate of a is given as [, [3, -7, -1], [-75, 27, 25]].
a⁻¹ Outcome
a⁻¹ = (1/302) * [, [3, -7, -1], [-75, 27, 25]].
Final Calculation (a⁻¹ * b)
Multiply a⁻¹ by the matrix b.
- (1/302) * [, [3, -7, -1], [-75, 27, 25]] * [, [-3],.
- (1/302) * [,, [-906]] is the product of the multiplication.
According to the audio, the outcome is (1/302) * [,, [-96]], which yields x = 600/302, y = 4/302, and z = -96/302.
Final Solution
As mentioned in the audio, the final solution is as follows: x = 2, y = 0, and z = -3.
