Problem 47 - Precalculus Mathematics for Calculus

Question

Use Cramer's Rule to solve the system. $$\left\\{\begin{aligned} x+y+z+w &=0 \\ 2 x &+w=0 \\ y-z &=0 \\ x &=1 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system does not have a unique solution; the coefficient matrix is singular.

Step 1: Write the system in matrix form

We can write the given system of equations in matrix form as follows. The system is:\[\begin{aligned}x + y + z + w &= 0 \2x + 0y + 0z + w &= 0 \0x + y - z + 0w &= 0 \x + 0y + 0z + 0w &= 1\end{aligned}\]In matrix form, this becomes:\[\begin{bmatrix}1 & 1 & 1 & 1 \2 & 0 & 0 & 1 \0 & 1 & -1 & 0 \1 & 0 & 0 & 0 \\end{bmatrix}\begin{bmatrix}x \y \z \w\end{bmatrix}=\begin{bmatrix}0 \0 \0 \1\end{bmatrix}\]

Step 2: Compute the determinant of the coefficient matrix

The next step in Cramer's Rule is to calculate the determinant of the coefficient matrix:\[A =\begin{bmatrix}1 & 1 & 1 & 1 \2 & 0 & 0 & 1 \0 & 1 & -1 & 0 \1 & 0 & 0 & 0 \\end{bmatrix}\]Calculate \( \det(A) \). By expanding along the fourth row, we have:\[\det(A) = 1\left|\begin{array}{ccc} 1 & 1 & 1 \ 2 & 0 & 0 \ 0 & 1 & -1 \end{array}\right| = 1(0 - 1(0)) = 0\]

Step 3: Analyzing the determinant

Since \( \det(A) = 0 \), the coefficient matrix is singular. A singular matrix means that the system may have no solution or infinitely many solutions. Therefore, Cramer's Rule is not applicable to find a unique solution for this system.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with a set of unknowns. The goal is to find the values of the unknowns that satisfy all the equations in the system at the same time. For example, the system you are dealing with here can be dissected as follows:

• Equation 1: The combined sum of variables \( x, y, z, \) and \( w \) equals zero.
• Equation 2: Two times \( x \) plus \( w \) must also equate to zero, meaning \( w = -2x \).
• Equation 3: The difference between \( y \) and \( z \) must be zero, meaning \( y = z \).
• Equation 4: \( x \) is defined outright as 1.

Systems can vary in complexity and the number they contain of unknowns and equations. They are often solved using methods such as substitution, elimination, or matrix operations like Cramer's Rule.

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements. It is a useful scalar that helps in understanding the properties of the matrix.

• For a 2x2 matrix, the determinant is found by subtracting the product of the diagonal from the product of the off-diagonal.
• For larger matrices, methods like cofactor expansion are used, making the calculation more complex.

In the context of Cramer's Rule, the determinant is crucial. If the determinant of the coefficient matrix is non-zero, the system has a unique solution. However, if it’s zero, like in the example, the system cannot be solved uniquely because it indicates the matrix is singular.

Singular Matrix

A singular matrix is a square matrix whose determinant is zero. This property dramatically affects the solutions to a system of equations.

• When a matrix is singular, it suggests that either no solution exists, or an infinite number of solutions are possible.
• This can occur due to the linear dependence of rows or columns, meaning one row or column can be expressed as a linear combination of the others.
• In the context of Cramer's Rule, the presence of a singular matrix means the rule is not applicable as it stages the basis for a unique solution.

Identifying a singular matrix is essential in determining the best method for solving a system since traditional methods depend on the matrix being non-singular for a guaranteed unique solution.