Problem 58 - Calculus for Biology and Medicine

Question

Suppose that $$ A=\left[\begin{array}{ll} a & 8 \\ 2 & 4 \end{array}\right], \quad X=\left[\begin{array}{l} x \\ y \end{array}\right], \quad \text { and } \quad B=\left[\begin{array}{l} b_{1} \\ b_{2} \end{array}\right] $$ (a) Show that when \(a \neq 4, A X=B\) has exactly one solution. (b) Suppose \(a=4\). Find conditions on \(b_{1}\) and \(b_{2}\) such that \(A X=B\) has (i) infinitely many solutions and (ii) no solutions. (c) Explain your results in (a) and (b) graphically.

Step-by-Step Solution

Answer

(a) Unique solution when \(a \neq 4\). (b) Infinite solutions if \(b_1 = b_2\), no solutions if \(b_1 \neq b_2\) when \(a = 4\).

Step 1: Formulate the Equation

We need to solve the matrix equation \( AX = B \) where \( A = \begin{bmatrix} a & 8 \ 2 & 4 \end{bmatrix} \), \( X = \begin{bmatrix} x \ y \end{bmatrix} \), and \( B = \begin{bmatrix} b_1 \ b_2 \end{bmatrix} \). This results in the system of equations: \( ax + 8y = b_1 \) and \( 2x + 4y = b_2 \).

Step 2: Analyze When a ≠ 4 for Unique Solution

For the system to have a unique solution, the determinant of matrix \( A \) should be non-zero. Compute the determinant: \( \text{det}(A) = a \cdot 4 - 8 \cdot 2 = 4a - 16 \). This is non-zero when \( 4a - 16 eq 0 \), or equivalently \( a eq 4 \).

Step 3: Analyze When a = 4 for Infinite or No Solutions

When \( a = 4 \), the determinant becomes zero, and the system may have either no solutions or infinitely many solutions. Substitute \( a = 4 \) into the equations: \( 4x + 8y = b_1 \) and \( 2x + 4y = b_2 \). The second equation can be simplified to \( x + 2y = \frac{b_2}{2} \). Both equations must be consistent for solutions to exist.

Step 4: Conditions for Infinitely Many Solutions

For infinitely many solutions when \( a = 4 \), the first equation must be a multiple of the second. Thus, \( 4x + 8y = 2(b_2/2) \) or \( b_1 = b_2 \). Then any \( x, y \) satisfying \( x + 2y = b_2/2 \) is a solution.

Step 5: Conditions for No Solutions

If there is no such consistency between the equations, specifically \( b_1 eq b_2 \), the system will have no solution, as the lines represented by the two equations will be parallel and distinct.

Step 6: Graphical Explanation

When \( a eq 4 \), matrix \( A \) corresponds to two lines that intersect at a single point, indicating a unique solution. For \( a = 4 \), the lines are either coincident (when \( b_1 = b_2 \)) leading to infinite solutions, or parallel (when \( b_1 eq b_2 \)) leading to no solutions.

Key Concepts

Determinant

The determinant of a matrix is a special number that can help us understand certain properties of that matrix. In the case of a 2x2 matrix like \( A \), the determinant is calculated by the formula: \[ \text{det}(A) = ad - bc \] where \( a \) and \( d \) are the elements on the main diagonal, and \( b \) and \( c \) are the other two elements.
For matrix \( A = \begin{bmatrix} a & 8 \ 2 & 4 \end{bmatrix} \), the determinant is:\[ \text{det}(A) = a \cdot 4 - 8 \cdot 2 = 4a - 16 \]
Why is the determinant important? It's a key factor in determining whether a system of equations has a unique solution. If the determinant is zero, the matrix is singular, meaning it may not have a single, unique solution, but rather none or infinitely many.
To summarize:

• If \( \text{det}(A) eq 0 \), the system can have a unique solution.
• If \( \text{det}(A) = 0 \), the system could have no solution or infinitely many solutions, depending on other factors.

Unique Solution

A unique solution in a system of equations occurs when exactly one set of values satisfies all equations simultaneously. In our exercise, we are dealing with matrix equations where a unique solution is guaranteed when certain conditions are met.
Let's break down why this happens:
1. **Non-Zero Determinant**: As mentioned earlier, the determinant of matrix \( A \) must not be zero. In this scenario, when \( a eq 4 \), the determinant \( 4a - 16 eq 0 \), ensuring that matrix \( A \) is invertible. This results in a distinct point of intersection for the lines represented by the equations.
2. **Intersection of Lines**: In the graphical representation of such systems, when the two lines intersect at exactly one point, this point corresponds to a unique solution for \( x \) and \( y \) in the matrix equation \( AX = B \).
3. **Invertibility**: A key result in linear algebra is that if a square matrix is invertible (non-zero determinant), then the system has exactly one solution. This is because we can multiply both sides of \( AX = B \) by the inverse of \( A \) to solve for \( X \).
Thus, ensuring \( A \) is not singular guarantees a unique solution.

Systems of Equations

At the heart of matrix equations is the idea of systems of linear equations. These systems can range from very simple to highly complex, depending on the number of equations and variables. Our scenario involves a 2x2 system:

• Equation 1: \( ax + 8y = b_1 \)
• Equation 2: \( 2x + 4y = b_2 \)

**Solving Systems**: To find solutions, the goal is to find values for \( x \) and \( y \) that satisfy both equations simultaneously. Here's how different determinant conditions affect these solutions:
- **Unique Solution**: As mentioned, this happens when \( a eq 4 \), as the lines intersect at a single point.
- **Infinite Solutions**: Occur when the lines coincide completely. For this to happen with \( a = 4 \), the equations should be multiples of each other. Thus, we need \( b_1 = b_2 \) so that both equations describe the same line.
- **No Solutions**: Happens when the lines are parallel yet distinct. This occurs when \( a = 4 \) and \( b_1 eq b_2 \), indicating the lines never meet.By understanding the properties of these equations, students can better master the concept of solving systems of equations using matrix methods.