Question

In each of the following, find (if possible) conditions on \(a, b,\) and \(c\) such that the system has no solution, one solution, or infinitely many solutions. a. \(\begin{aligned} 3 x+y-z &=a \\ x-y+2 z &=b \\ 5 x+3 y-4 z &=c \end{aligned}\) b. \(\begin{aligned} 2 x+y-z &=a \\ 2 y+3 z &=b \\ x \quad-z &=c \end{aligned}\) c. \(\begin{aligned}-x+3 y+2 z &=-8 \\ x+z &=2 \\ 3 x+3 y+a z &=b \end{aligned}\) d. \(\begin{aligned} & x+a y=0 \\ & y+b z=0 \\ & z+c x=0 \end{aligned}\) e. \(\begin{aligned} 3 x-y+2 z &=3 \\ x+y-z &=2 \\ 2 x-2 y+3 z &=b \end{aligned}\) \(\begin{array}{rrrr}\text { f. } & x+\quad a y- & z & =1 \\ & -x+(a-2) y+ & z & =-1 \\ & 2 x+ & 2 y+(a-2) & z= & 1\end{array}\)

Short answer

Each subpart has unique conditions; analyze each matrix for uniqueness or dependency conditions.

Step-by-step solution

Analyze the given system of equations (a)

Consider the following system of equations:\[\begin{aligned} 3x+y-z &= a \ x-y+2z &= b \ 5x+3y-4z &= c \end{aligned}\]To solve this, we need to examine conditions where this system might be inconsistent (no solution), have a unique solution (one solution), or dependent solutions (infinitely many solutions).

Determine the determinant for system (a)

For uniqueness or dependency, examine the determinant of the coefficient matrix.The coefficient matrix is:\[\begin{bmatrix} 3 & 1 & -1 \ 1 & -1 & 2 \ 5 & 3 & -4 \end{bmatrix}\]Calculate the determinant:\[\text{det} = 3(-1)(-4) + 1(2)(5) + (-1)(1)(3) - (-1)(1)(5) - 2(1)(5) - 3(3)(1) = -14 \]This is nonzero, indicating the system has a unique solution if no condition on \(a\), \(b\), \(c\) is specified. For no solution or infinitely many solutions, the rank of augmented matrix and coefficient matrix should be examined with specific \(a\), \(b\), and \(c\).

Analyze specified solutions for system (b)

Consider:\[\begin{aligned} 2x + y - z &= a \ 2y + 3z &= b \ x - z &= c \end{aligned}\]The coefficient matrix is:\[\begin{bmatrix} 2 & 1 & -1 \ 0 & 2 & 3 \ 1 & 0 & -1 \end{bmatrix}\]Calculate the determinant:\[\text{det} = 2(2)(-1) + 1(3)(1) + (-1)(0)(0) - (-1)(2)(1) - 3(1)(2) - 2(0)(1) = -1 \]Non-zero determinant implies a unique solution. Check for dependency or inconsistency by analyzing augmented matrix: if any linearly dependent or contradictions arise from given \(a, b, c\).

System's possible solutions for (c)

Given system:\[\begin{aligned} -x + 3y + 2z &= -8 \ x + z &= 2 \ 3x + 3y + az &= b \end{aligned}\]Coefficient matrix:\[\begin{bmatrix} -1 & 3 & 2 \ 1 & 0 & 1 \ 3 & 3 & a \end{bmatrix}\]Calculate its determinant:\[\text{det} = -1(0)(a) + 3(1)(3) + 2(3)(3) - (2)(0)(3) - (3a)(3) - (-1)(1) = 3a - 24 \]Set dependent (for infinitely many solutions) if the determinant \(3a - 24 = 0\). if any specific solutions exist stating by conflicting augmented matrix values.

Enforce conditions for system (d)

Examine the system:\[\begin{aligned} & x + ay = 0 \ & y + bz = 0 \ & z + cx = 0 \end{aligned}\]This system's coefficient matrix is:\[\begin{bmatrix} 1 & a & 0 \ 0 & 1 & b \ c & 0 & 1 \end{bmatrix}\]Determinant is:1. For \((a, b, c) = (0, 0, 0)\) means trivial solutions only.Non-zero determinant leads to all zeros.Unique only if coefficients diverge from such degeneracy; hence each value combination implies no solution, unique or infinite as an indirectly dependent structure.

Investigate conditions for system (e)

Given:\[\begin{aligned} 3x - y + 2z &= 3 \ x + y - z &= 2 \ 2x - 2y + 3z &= b \end{aligned}\]Calculate determinant of\[\begin{bmatrix} 3 & -1 & 2 \ 1 & 1 & -1 \ 2 & -2 & 3 \end{bmatrix}\]Find determinant value which determines uniqueness. Compare ranks between coefficient versus augmented matrices by \(b\) either stabilizing or conflicting.

Derive potential solutions for system form (f)

Consider:\[\begin{array}{rrrr}x + & ay - & z &= 1 \ -x + &(a-2)y + & z &= -1 \ 2x + & 2y + &(a-2)z &= 1 \end{array}\]Formulate determinants for solutions:\[\begin{bmatrix} 1 & a & -1 \ -1 & a-2 & 1 \ 2 & 2 & a-2 \end{bmatrix}\]Feature determinant consistency: \((a-6)(a)= 0\), leading solutions uniquely or infinitely.Ensure rank confirms when consistent versus inconsistent checks.

Key concepts

Determinants

Understanding determinants is critical when solving systems of linear equations. In a nutshell, the determinant of a matrix can help us find out whether a system of linear equations has a solution, and if so, how many solutions it has.

Consider a coefficient matrix derived from linear equations. If this matrix is square, which means it has the same number of rows and columns, you can calculate its determinant. The calculation involves a specific arithmetic operation that reflects the matrix's orientation in space. For a 3x3 matrix, this is done using a combination of additions and products of elements in specific patterns.

If the determinant is non-zero, the system has a unique solution. This indicates that the planes intersect at exactly one point. Conversely, if the determinant is zero, the system might have either no solutions or infinitely many solutions, depending on the relations between the equations.

Matrix Rank

The rank of a matrix is another crucial concept in understanding systems of linear equations. Put simply, the rank is the number of linearly independent rows or columns in the matrix. It provides insight into the solutions' existence and nature.

- For a system of equations to have a unique solution, the rank of the coefficient matrix should equal the number of variables. - If the rank is less than the number of variables, the system might be dependent (infinitely many solutions) or inconsistent (no solutions), depending on the augmented matrix's rank.

When dealing with augmented matrices—which include the coefficient matrix and an additional column from the constants in the equations—you compare the rank of the coefficient matrix with the augmented matrix. If both ranks are equal but less than the number of variables, there are infinitely many solutions. If the ranks differ, the system has no solution.

Solutions of Linear Systems

The solutions to a system of linear equations can be categorized as follows: no solution, one solution, or infinitely many solutions.

For a system to have no solution, it generally means the equations represent parallel planes that never intersect. This condition often occurs when the ranks of the coefficient matrix and the augmented matrix do not match.

If there is exactly one solution, the linear equations represent intersecting planes crossing at a single point. This is the typical case when the determinant of the coefficient matrix is non-zero, and the ranks of both matrices are equal to the number of variables.

Infinitely many solutions arise when the equations represent planes that intersect along a line or when they are essentially the same plane repeated multiple times. This condition is observed when the ranks of both the coefficient and augmented matrices are the same but less than the number of variables, corresponding to a zero determinant in a square system.