Math Problem Statement

find the value of β€œa” such that the following system of linear

equations

𝒙 + π’š + 𝒂𝒛 = πŸ‘, 𝒙 + π’‚π’š + 𝒛 = 𝟐, 𝒂𝒙 + π’š βˆ’ 𝒛 = 𝟏

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, we’ll 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:

  1. x+y+az=3x + y + az = 3
  2. x+ay+z=2x + ay + z = 2
  3. ax+yβˆ’z=1ax + y - z = 1

In matrix form, this can be written as: (11a1a1a1βˆ’1)(xyz)=(321)\begin{pmatrix} 1 & 1 & a \\ 1 & a & 1 \\ a & 1 & -1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \\ 1 \end{pmatrix}

To determine the solution types, we’ll examine the determinant of the coefficient matrix.

Step 1: Calculate the Determinant

The determinant DD of the matrix is: D=∣11a1a1a1βˆ’1∣D = \begin{vmatrix} 1 & 1 & a \\ 1 & a & 1 \\ a & 1 & -1 \end{vmatrix}

Expanding along the first row, we get: D=1β‹…βˆ£a11βˆ’1βˆ£βˆ’1β‹…βˆ£11aβˆ’1∣+aβ‹…βˆ£1aa1∣D = 1 \cdot \begin{vmatrix} a & 1 \\ 1 & -1 \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & 1 \\ a & -1 \end{vmatrix} + a \cdot \begin{vmatrix} 1 & a \\ a & 1 \end{vmatrix}

Calculating each minor:

  1. ∣a11βˆ’1∣=a(βˆ’1)βˆ’1β‹…1=βˆ’aβˆ’1=βˆ’(a+1)\begin{vmatrix} a & 1 \\ 1 & -1 \end{vmatrix} = a(-1) - 1 \cdot 1 = -a - 1 = -(a + 1)
  2. ∣11aβˆ’1∣=1(βˆ’1)βˆ’1β‹…a=βˆ’1βˆ’a=βˆ’(1+a)\begin{vmatrix} 1 & 1 \\ a & -1 \end{vmatrix} = 1(-1) - 1 \cdot a = -1 - a = -(1 + a)
  3. ∣1aa1∣=1β‹…1βˆ’aβ‹…a=1βˆ’a2\begin{vmatrix} 1 & a \\ a & 1 \end{vmatrix} = 1 \cdot 1 - a \cdot a = 1 - a^2

Thus: D=1β‹…(βˆ’(a+1))βˆ’1β‹…(βˆ’(1+a))+aβ‹…(1βˆ’a2)D = 1 \cdot (-(a + 1)) - 1 \cdot (-(1 + a)) + a \cdot (1 - a^2) D=βˆ’(a+1)+(1+a)+a(1βˆ’a2)D = -(a + 1) + (1 + a) + a(1 - a^2) D=0+a(1βˆ’a2)D = 0 + a(1 - a^2)

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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 βˆ’ fh) βˆ’ b(di βˆ’ fg) + c(dh βˆ’ 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)