Math Problem Statement
find the value of βaβ such that the following system of linear
equations
π + π + ππ = π, π + ππ + π = π, ππ + π β π = π
have
i) exactly one solution
ii) No solution
iii) Infinite number of solutions and find them
Solution
To analyze the conditions for the system of equations to have exactly one solution, no solution, or infinitely many solutions, weβll start by representing the system in matrix form and determining when it is consistent, consistent with a unique solution, or inconsistent.
System of Equations
The system is:
In matrix form, this can be written as:
To determine the solution types, weβll examine the determinant of the coefficient matrix.
Step 1: Calculate the Determinant
The determinant of the matrix is:
Expanding along the first row, we get:
Calculating each minor:
Thus:
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Matrix Determinants
Consistency of Equations
Formulas
Determinant of 3x3 matrix D = |a b c; d e f; g h i| = a(ei β fh) β b(di β fg) + c(dh β eg)
Conditions for solution types in systems of equations: If D β 0, there is a unique solution; if D = 0, further analysis is needed for no or infinite solutions.
Theorems
Determinant properties for system consistency
Matrix rank and consistency of linear systems
Suitable Grade Level
Undergraduate Level (or Advanced High School)
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