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. \(A=\left[\begin{array}{cc}0 & -6 \\ 9 & -10\end{array}\right], \quad \vec{b}=\left[\begin{array}{c}6 \\ -17\end{array}\right]\)

Short answer

The solution is \(\vec{x} = \begin{bmatrix} -3 \\ -1 \end{bmatrix}\).

Step-by-step solution

Calculate the Determinant of Matrix A

The determinant of a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) is given by \(\operatorname{det}(A) = ad - bc\). For the matrix \(A = \begin{bmatrix} 0 & -6 \ 9 & -10 \end{bmatrix}\), compute the determinant as follows:\[\operatorname{det}(A) = (0)(-10) - (9)(-6) = 0 + 54 = 54\]

Create and Calculate Determinants of Matrices A_i

For each \(i\), replace the \(i\)-th column of \(A\) with \(\vec{b}\) and calculate the determinant of \(A_i\).- For \(i = 1\), replace the first column:\[A_1 = \begin{bmatrix} 6 & -6 \ -17 & -10 \end{bmatrix}\]\[\operatorname{det}(A_1) = (6)(-10) - (-6)(-17) = -60 - 102 = -162\]- For \(i = 2\), replace the second column:\[A_2 = \begin{bmatrix} 0 & 6 \ 9 & -17 \end{bmatrix}\]\[\operatorname{det}(A_2) = (0)(-17) - (6)(9) = 0 - 54 = -54\]

Check for the Applicability of Cramer's Rule

Cramer's Rule can be used if \(\operatorname{det}(A) eq 0\). Since \(\operatorname{det}(A) = 54 eq 0\), Cramer's Rule can be applied to solve \(A \vec{x} = \vec{b}\).

Solving for Unknowns using Cramer's Rule

Using Cramer's Rule, the solutions for the components of \(\vec{x}\) are given by:\[x_1 = \frac{\operatorname{det}(A_1)}{\operatorname{det}(A)} = \frac{-162}{54} = -3\]\[x_2 = \frac{\operatorname{det}(A_2)}{\operatorname{det}(A)} = \frac{-54}{54} = -1\]

Solution Interpretation

The solution to the system \(A \vec{x} = \vec{b}\) is \(\vec{x} = \begin{bmatrix} -3 \ -1 \end{bmatrix}\). Since the determinant of \(A\) is non-zero, the system has a unique solution.

Key concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can tell us a lot about the matrix itself. For a 2x2 matrix, the determinant is calculated using the formula:\[\operatorname{det}(A) = ad - bc\]where \(A\) is the matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\). The determinant can help us determine whether a matrix is invertible or not, among other properties. In this context, the determinant plays a critical role in solving linear systems using methods like Cramer's Rule.In our example with matrix \(A = \begin{bmatrix} 0 & -6 \ 9 & -10 \end{bmatrix}\), the determinant is computed as follows:\[\operatorname{det}(A) = 0(-10) - (-6)(9) = 54\]This tells us that matrix \(A\) is invertible and can be used with Cramer's Rule to solve linear equations.

Matrix Equation

A matrix equation is a compact way of representing a system of linear equations. In matrix form, these equations use a coefficient matrix \(A\), a variable vector \(\vec{x}\), and a constant vector \(\vec{b}\), summarized as:\[A \vec{x} = \vec{b}\]Here, \(A\) is a square matrix containing the coefficients of the system, \(\vec{x}\) is the vector of variables, and \(\vec{b}\) represents the constants on the other side of the equations.Writing the system in matrix form allows us to utilize efficient computational tools and mathematical methods, like matrix inversion and determinants, to find solutions. It simplifies complex systems, letting us use matrix operations to solve equations that would otherwise be cumbersome. In our exercise, the equation \(A \vec{x} = \vec{b}\) is:\[\begin{bmatrix} 0 & -6 \ 9 & -10 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} 6 \ -17 \end{bmatrix}\]

System of Linear Equations

A system of linear equations is a collection of linear equations involving the same set of variables. Solving these systems involves finding the values for the variables that make all equations true simultaneously.In a practical scenario, a system like this can describe various real-world situations such as electrical circuits, economics, or physics problems. The equations generally form a line when graphed, and the solution to the system is where these lines intersect on a plane.In our exercise, we deal with a system which is represented by two equations:- \(0x_1 - 6x_2 = 6\)- \(9x_1 - 10x_2 = -17\)One of the popular methods to solve such systems is Cramer's Rule, which uses determinants to find the solution when determinants are non-zero, indicating that a unique solution exists. The use of the matrix equation \(A \vec{x} = \vec{b}\) allows us to apply Cramer's Rule effectively.