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) One solution if \(a \neq 4\). (b) Infinitely many if \(b_1 = 8\), \(b_2 = 4\); none otherwise. (c) Unique solution if lines intersect once, infinite if they coincide, none if parallel.

Step 1: Matrix Equation and Determinant

Consider the equation \(AX = B\). For us to determine the solutions, we check the determinant of matrix \(A\). If the determinant is non-zero, the system has a unique solution; if the determinant is zero, we'll explore the possibilities of no solution or infinitely many solutions.The determinant of \(A\) is \(|A| = a \cdot 4 - 8 \cdot 2 = 4a - 16\). Therefore, the determinant is \(|A| = 4(a - 4)\).

Step 2: Analyzing Part (a)

For the matrix equation \(AX = B\) to have exactly one solution, the determinant of \(A\) should be non-zero. From the previous step, this happens when \(4(a - 4) eq 0\), which simplifies to \(a eq 4\). This means the matrix \(A\) is invertible and there exists exactly one solution.

Step 3: Analyzing Part (b) when \(a=4\)

When \(a = 4\), the determinant becomes zero, \(4(4 - 4) = 0\), implying the matrix \(A\) is not invertible, and special conditions must be met for solutions.If \(b_1 = 8 \) and \(b_2 = 4\), the system has infinitely many solutions because the determinant condition will not affect the ability to satisfy the equation in multiple ways.If \(b_1 eq 8 \) or \(b_2 eq 4\), the right-hand side cannot be a linear combination of the columns of \(A\), resulting in no solutions.

Step 4: Graphical Explanation

For part (a), when \(a eq 4\), the matrix \(A\) defines two lines that intersect at exactly one point, corresponding to the unique solution.For part (b) with \(a = 4\), the lines coincide if \(b_1 = 8\) and \(b_2 = 4\), representing infinitely many solutions as the entire line is a solution. Otherwise, the lines are parallel without intersections, corresponding to no solutions.

Key Concepts

Matrix Equations

A matrix equation is a mathematical representation where matrices and vectors are used to describe linear relationships. For example, in our exercise, the equation \(AX = B\) involves multiplying matrix \(A\) by column vector \(X\) to yield another column vector \(B\). This is analogous to solving systems of linear equations. Each row of the matrix equation corresponds to a linear equation involving variables from the vector \(X\).

Matrix multiplication follows specific rules, which dictate how rows of the first matrix interact with columns of the second. It's essential:

• The number of columns in \(A\) must match the number of rows in \(X\).
• Each element in the resulting vector \(B\) is calculated as a dot product of a row in \(A\) and the vector \(X\).

Understanding these basics is crucial to approach more complex operations like finding solutions or manipulating matrix equations.

Determinant

The determinant provides a single value that offers insights into a matrix's properties. For a 2x2 matrix like \(A\) in our exercise, the determinant is calculated as \( \text{det}(A) = a_{11} \cdot a_{22} - a_{12} \cdot a_{21} \), where \(a_{11}, a_{12}, a_{21},\) and \(a_{22}\) are elements of \(A\). This helps us understand whether the matrix equations have a unique solution, no solution, or infinitely many solutions.

If the determinant is zero, the matrix is singular, meaning it cannot be inverted, and the linear system may lack a unique solution. For non-zero determinants, the matrix is invertible, guaranteeing a single unique solution. In our example, the determinant is calculated as \(|A| = 4(a - 4)\). Hence, if \(a eq 4\), \(|A|eq 0\), leading to a unique solution.

Matrix Invertibility

Matrix invertibility is a critical concept in determining the nature of solutions to matrix equations. A matrix is invertible if it has an inverse, a condition that is only possible when the determinant is non-zero. For the matrix \(A\) in our exercise, the invertibility depends on this determinant.

When \(a eq 4\), the determinant \(4(a - 4)\) is non-zero, making \(A\) invertible. Invertibility implies that we can find a matrix \(A^{-1}\) such that \(A \cdot A^{-1} = I\), where \(I\) is the identity matrix. This property ensures the matrix equation \(AX = B\) can be rewritten as \(X = A^{-1}B\), resulting in a unique solution.

Conversely, if \(a = 4\), the determinant becomes zero and \(A\) is non-invertible, so the system of equations may either be inconsistent (no solutions) or dependent (infinitely many solutions) based on the values of \(b_1\) and \(b_2\) in vector \(B\).