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. \(A=\left[\begin{array}{cc}-8 & 16 \\ 10 & -20\end{array}\right], \quad \vec{b}=\left[\begin{array}{c}-48 \\ 60\end{array}\right]\)

Short answer

The determinant of A is 0, and Cramer's Rule cannot be used. The system likely has infinitely many solutions.

Step-by-step solution

Calculate the Determinant of A

Given the matrix \(A = \begin{bmatrix} -8 & 16 \ 10 & -20 \end{bmatrix}\), we calculate the determinant using the formula for a 2x2 matrix: \[ \det(A) = ad - bc \]where \(a = -8\), \(b = 16\), \(c = 10\), and \(d = -20\). Thus,\[ \det(A) = (-8)(-20) - (16)(10) = 160 - 160 = 0 \]

Determine the Use of Cramer's Rule

Cramer's Rule can only be applied if \(\det(A) eq 0\). From Step 1, we have \(\det(A) = 0\). This implies that Cramer's Rule cannot be used.

Analyze Existence of Solutions

Since \(\det(A) = 0\), matrix \(A\) is singular, indicating that it does not have an inverse. This generally means the system either has no solutions or has infinitely many solutions. We need additional analysis to determine which is the case. However, the problem requires stating the existence of solutions when Cramer's Rule cannot be used. Given the structure, it is likely there are infinitely many solutions.

Key concepts

Understanding the Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from a square matrix. It gives important information about the matrix, specifically its invertibility. For a 2x2 matrix like\[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]the formula to find the determinant is simple:\[\det(A) = ad - bc\]
So, for matrix\[A = \begin{bmatrix} -8 & 16 \ 10 & -20 \end{bmatrix}\]we find:\[\det(A) = (-8)(-20) - (16)(10) = 160 - 160 = 0\]
When the determinant equals zero, the matrix is said to be singular, which leads us to some crucial implications for solving linear equations using Cramer's Rule.

What is a Singular Matrix?

A singular matrix is one for which the determinant is zero. This characteristic has important consequences:

• It does not have an inverse.
• It may lead to a system of equations having no solutions or infinitely many solutions.

In our example, since\(\det(A) = 0\)the matrix\(A\)is singular. This information tells us that standard methods like Cramer's Rule cannot be used because they require the determinant to be non-zero. Instead, we must consider the possibility of the system having infinitely many solutions or possibly no solutions at all.

Infinitely Many Solutions in Linear Equations

When we discover that a matrix is singular, one common outcome is the presence of infinitely many solutions. This happens because the matrix has dependent rows or columns, meaning they do not add new information to solve the equation uniquely.
In the context of our calculation of\(\det(A)\), we established that Cramer's Rule cannot apply. However, the structure of matrix\(A\) suggests that the linear equation system\(A\vec{x} = \vec{b}\)may indeed have infinitely many solutions.
If the vectors connected by the equation are parallel or share linear dependencies, the solution vector\(\vec{x}\)can indeed take many forms, illustrating one of the fascinating aspects of linear algebra.