Question

Matrices \(A\) and \(\vec{b}\) are given. (a) Give \(\operatorname{det}(A)\) and \(\operatorname{det}\left(A_{i}\right)\) for all \(i\). (b) Use Cramer's Rule to solve \(A \vec{x}=\vec{b}\). If Cramer's Rule cannot be used to find the solution, then state whether or not a solution exists. \(A=\left[\begin{array}{cc}7 & 14 \\ -2 & -4\end{array}\right], \quad \vec{b}=\left[\begin{array}{c}-1 \\ 4\end{array}\right]\)

Short answer

Determinant is 0; Cramer's Rule cannot be applied.

Step-by-step solution

Calculate Determinant of Matrix A

The determinant of a 2x2 matrix \( A = \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \) is calculated using the formula \( \operatorname{det}(A) = ad - bc \). For matrix \( A = \left[ \begin{array}{cc} 7 & 14 \ -2 & -4 \end{array} \right] \), we substitute the values into the formula: \( \operatorname{det}(A) = (7)(-4) - (14)(-2) = -28 + 28 = 0 \). Thus, \( \operatorname{det}(A) = 0 \).

Assess Cramer's Rule Applicability

Cramer's Rule can only be applied if the determinant of matrix \( A \) is non-zero. Since \( \operatorname{det}(A) = 0 \), Cramer's Rule cannot be used to find a unique solution to the system \( A \vec{x} = \vec{b} \). This means that the system either has no solutions or infinitely many solutions.

Key concepts

Determinant of a Matrix

Understanding the determinant of a matrix is crucial in linear algebra. It is a special number that can be calculated from a square matrix. For a 2x2 matrix, the determinant tells us about the properties of the matrix, like whether it is invertible or not. The formula for the determinant of a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\)is \(\operatorname{det}(A) = ad - bc\).
In our example, matrix \(A = \begin{bmatrix} 7 & 14 \ -2 & -4 \end{bmatrix}\), the determinant is calculated as follows:

• Multiply the elements from the main diagonal: \(7 \times (-4)\).
• Multiply the elements from the other diagonal: \(14 \times (-2)\).
• Subtract the second product from the first: \((-28) - (-28) = 0\).

The zero value of the determinant indicates that the matrix does not have an inverse and systems of equations involving this matrix may not have unique solutions.

Matrix Equations

Matrix equations are like regular equations but involve matrices. They often look like \(A \vec{x} = \vec{b}\), where \(A\) is a matrix, \(\vec{x}\) is a column vector of variables, and \(\vec{b}\) is a column vector of constants. Solving these equations involves finding the values of \(\vec{x}\) such that the equation holds true.
When solving a matrix equation using techniques like Cramer's Rule, calculating the determinant of the matrix \(A\) is vital. If \(\operatorname{det}(A) = 0\), as in our exercise, it suggests that the matrix equation may not have a unique solution or might have no solution at all. This happens because the matrix can't be inverted to solve the equation using Cramer's Rule. Therefore, the system of equations represented by the matrix equation \(A \vec{x} = \vec{b}\) is either inconsistent or dependent, meaning it could have infinitely many solutions.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, matrices, and linear transformations. It serves as a foundational framework for understanding systems of linear equations and linear mappings between vector spaces.
A key goal in linear algebra is to solve systems of linear equations, which can often be represented succinctly in matrix form. Matrix algebra provides tools like determinants, inverses, and methods such as Cramer's Rule to find solutions to these systems.
However, not all systems can be solved using these methods. In cases like our exercise, where the determinant of the matrix is zero, linear algebra offers insights into why solutions might not be unique or existent. These concepts have vast applications in numerous fields such as engineering, computer science, physics, and more, allowing for the modeling and solving of complex real-world problems through a clear mathematical structure.