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. \(A=\left[\begin{array}{cc}9 & 5 \\ -4 & -7\end{array}\right], \quad \vec{b}=\left[\begin{array}{c}-45 \\ 20\end{array}\right]\)

Short answer

Solution: \( \vec{x} = \begin{bmatrix} -5 \\ 0 \end{bmatrix} \).

Step-by-step solution

Compute the Determinant of Matrix A

Use the formula for the determinant of a 2x2 matrix: \( \operatorname{det}(A) = ad - bc \). For matrix \( A = \begin{bmatrix} 9 & 5 \ -4 & -7 \end{bmatrix} \), calculate \( \operatorname{det}(A) = (9)(-7) - (5)(-4) \). This simplifies to \( -63 + 20 = -43 \). Thus, \( \operatorname{det}(A) = -43 \).

Compute the Determinants of Matrices A_i

Using matrices \( A_1 \) and \( A_2 \) generated by replacing each column of \( A \) with \( \vec{b} \), calculate their determinants:1. \( A_1 = \begin{bmatrix} -45 & 5 \ 20 & -7 \end{bmatrix} \), and calculate: \( \operatorname{det}(A_1) = (-45)(-7) - (5)(20) = 315 - 100 = 215 \).2. \( A_2 = \begin{bmatrix} 9 & -45 \ -4 & 20 \end{bmatrix} \), and calculate: \( \operatorname{det}(A_2) = (9)(20) - (-45)(-4) = 180 - 180 = 0 \).

Use Cramer's Rule to Solve for \(\vec{x}\)

Since \( \operatorname{det}(A) eq 0 \), Cramer's Rule can be applied. According to Cramer's Rule:- \( x_1 = \frac{\operatorname{det}(A_1)}{\operatorname{det}(A)} = \frac{215}{-43} = -5 \).- \( x_2 = \frac{\operatorname{det}(A_2)}{\operatorname{det}(A)} = \frac{0}{-43} = 0 \).Thus, the solution \( \vec{x} = \begin{bmatrix} -5 \ 0 \end{bmatrix} \).

Key concepts

Determinant

The determinant is a special number that can be calculated from a square matrix. It provides important properties of the matrix, such as invertibility. For a 2x2 matrix, the determinant can be easily found using the formula \( \operatorname{det}(A) = ad - bc \), where \(a\), \(b\), \(c\), and \(d\) are elements of the matrix.
For example, consider matrix \(A = \begin{bmatrix} 9 & 5 \ -4 & -7 \end{bmatrix} \). To find \( \operatorname{det}(A) \), multiply 9 by -7 and subtract the product of 5 and -4: \( 9(-7) - 5(-4) = -63 + 20 = -43 \).
The determinant of \(A\) is \(-43\). This non-zero result tells us that matrix \(A\) is invertible and we can use certain methods, like Cramer's Rule, to solve systems of equations.

2x2 Matrix

A 2x2 matrix is a simple form of a matrix with 2 rows and 2 columns. It is often used in linear algebra to solve systems of linear equations. The general form of a 2x2 matrix is \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), where \(a\), \(b\), \(c\), and \(d\) are the matrix elements.
Working with a 2x2 matrix is straightforward due to the limited size, making calculations like determinants and matrix solutions more manageable. Their compact nature makes understanding foundational concepts in linear algebra easier.
For example, you can compute the determinant or perform matrix multiplication easily using these small matrices, which are foundational skills for understanding more complex matrix operations and systems.

Linear Algebra

Linear algebra is a branch of mathematics focusing on vectors, matrices, and linear transformations. It plays a crucial role in understanding systems of linear equations, especially with matrices like 2x2 matrices.
In linear algebra, matrices are used to represent and solve systems of equations. This involves finding solutions that satisfy all equations in the system simultaneously. Methods like Cramer's Rule and the use of determinants arise from linear algebra to effectively solve these problems.
Linear algebra is essential not just in mathematics, but in fields like computer science, physics, and engineering. It helps in data representation, computer graphics, and understanding mathematical models in sciences.

Matrix Solution

Matrix solutions refer to solving systems of equations using matrix operations. One powerful tool in linear algebra is Cramer's Rule, which can solve equations of the form \(A \vec{x} = \vec{b}\).
When dealing with a 2x2 matrix, ensure that the determinant of the matrix \(A\) is non-zero. If it is, Cramer's Rule allows the solution to be found by replacing the columns of \(A\) with the resulting vector \(\vec{b}\) and calculating determinants.

• Replace the first column with \(\vec{b}\) to get matrix \(A_1\).
• Replace the second column of \(A\) with \(\vec{b}\) to get matrix \(A_2\).

The solutions for \(\vec{x}\) are found using \( x_i = \frac{\operatorname{det}(A_i)}{\operatorname{det}(A)} \). This approach is direct and efficient for small systems, ensuring solutions are accurate when applied correctly.