Problem 29 - Differential Equations and Linear Algebra

Question

Consider the system of linear equations $$ \begin{array}{l} a_{11} x_{1}+a_{12} x_{2}=b_{1} \\ a_{21} x_{1}+a_{22} x_{2}=b_{2} \end{array} $$ Define \(\Delta, \Delta_{1},\) and \(\Delta_{2}\) by $$ \Delta=a_{11} a_{22}-a_{12} a_{21} $$ $$ \Delta_{1}=a_{22} b_{1}-a_{12} b_{2}, \Delta_{2}=a_{11} b_{2}-a_{12} b_{1} $$ (a) Show that the given system has a unique solution if and only if \(\Delta \neq 0,\) and that the unique solution in this case is \(x_{1}=\Delta_{1} / \Delta, x_{2}=\Delta_{2} / \Delta\) (b) If \(\Delta=0\) and \(a_{11} \neq 0,\) determine the conditions on \(\Delta_{2}\) that would guarantee that the system has (i) no solution, (ii) an infinite number of solutions. (c) Interpret your results in terms of intersections of straight lines.

Step-by-Step Solution

Answer

In summary, the given system of linear equations has a unique solution if \(\Delta \neq 0\) and the solution is given as \(x_1 = \frac{\Delta_1}{\Delta}\) and \(x_2 = \frac{\Delta_2}{\Delta}\). If \(\Delta = 0\) and \(a_{11} \neq 0\), the system has no solution if \(\Delta_2 \neq 0\) and an infinite number of solutions if \(\Delta_2 = 0\). In the context of intersections of straight lines, a unique solution corresponds to the lines intersecting at a single point, no solution corresponds to parallel lines, and infinite solutions correspond to coincident lines.

Step 1: (a) Determine unique solution conditions

First, let's demonstrate that the system has a unique solution if and only if \(\Delta \neq 0\). Then, we will express the solution in terms of \(\Delta_1\) and \(\Delta_2\). We can write the system in matrix form: \[ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} = \begin{pmatrix} b_{1} \\ b_{2} \end{pmatrix}.\] Notice that \(\Delta\) is the determinant of the coefficient matrix. If \(\Delta \neq 0\), we know that the matrix is invertible. Therefore, we can find the inverse matrix \(A^{-1}\) and multiply both sides of the equation by \(A^{-1}\) to obtain the unique solution: \[ \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} = A^{-1} \begin{pmatrix} b_{1} \\ b_{2} \end{pmatrix}.\] Now, let's find the inverse matrix and multiply it by the right-hand side to obtain the solution: \[A^{-1} = \frac{1}{\Delta}\begin{pmatrix} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{pmatrix},\] \[ \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} = \frac{1}{\Delta} \begin{pmatrix} a_{22} & -a_{12} \\-a_{21} & a_{11} \end{pmatrix}\begin{pmatrix} b_{1} \\ b_{2} \end{pmatrix} = \frac{1}{\Delta} \begin{pmatrix} a_{22}b_{1} - a_{12}b_{2} \\ -a_{21}b_{1} + a_{11}b_{2} \end{pmatrix}.\] Comparing with the definitions of \(\Delta_1\) and \(\Delta_2\), we get the unique solution: \[x_1 = \frac{\Delta_1}{\Delta}, \quad x_2 = \frac{\Delta_2}{\Delta}.\]

Step 2: (b) Determine no solution and infinite solutions conditions

In this part, we are given that \(\Delta = 0\) and \(a_{11} \neq 0\). We want to find conditions on \(\Delta_2\) that guarantee either no solution or an infinite number of solutions. First, notice that since \(\Delta = a_{11} a_{22} - a_{12} a_{21} = 0\), we have \(a_{22} = \frac{a_{12}a_{21}}{a_{11}}\). Now, we rewrite the second equation, substituting \(a_{22}\): \[a_{21}x_1 + \frac{a_{12}a_{21}}{a_{11}}x_2 = b_2.\] Now, let's solve the first equation for \(x_1\): \[x_1 = \frac{b_{1} - a_{12}x_2}{a_{11}}.\] Substitute this expression for \(x_1\) in the modified second equation: \[\frac{a_{21}(b_1 - a_{12}x_2)}{a_{11}}+\frac{a_{12}a_{21}}{a_{11}}x_2 = b_2.\] Simplify the equation: \[a_{21}b_1 - a_{12}a_{21}x_2 + a_{12}a_{21}x_2 = a_{11}b_2.\] From this equation, we can see the conditions on \(\Delta_2\): (i) No solution (\(x_1\) and \(x_2\) are undefined) if \(a_{21}b_1 \neq a_{11}b_2\), or \(\Delta_2 \neq 0\). (ii) Infinite solutions (the system is equivalent) if \(a_{21}b_1 = a_{11}b_2\), or \(\Delta_2 = 0\).

Step 3: (c) Interpret results for intersection of lines

Finally, let's interpret our results in terms of intersections of straight lines. The given system of linear equations can be represented as a pair of lines in a coordinate plane: \[y_1 = \frac{-a_{11}}{a_{12}}x_1 + \frac{b_1}{a_{12}},\] \[y_2 = \frac{-a_{21}}{a_{22}}x_2 + \frac{b_2}{a_{22}}.\] The system has a unique solution if and only if the coefficient determinant \(\Delta \neq 0\). In this case, both lines have different slopes, and they intersect at a single point. This point represents the unique solution \((x_1, x_2)\). When \(\Delta = 0\), the system could have either no solution or an infinite number of solutions. This means the lines are either parallel (no intersection) or coincident (infinite intersections). We found conditions on \(\Delta_2\) for these cases: (i) No solution (\(\Delta_2 \neq 0\)): The lines are parallel and distinct, so they never intersect. This means there is no pair \((x_1, x_2)\) that satisfies both equations. (ii) Infinite solutions (\(\Delta_2 = 0\)): The lines coincide, so every point on one line is also a point on the other line. Every pair \((x_1, x_2)\) on the line satisfies both equations.

Key Concepts

Determinants

In the context of systems of linear equations, determinants play a crucial role. For a system of equations expressed in matrix form, the determinant of the coefficient matrix helps us understand various properties of the system. Specifically, the determinant, denoted as \( \Delta \), is computed as \( a_{11} a_{22} - a_{12} a_{21} \). This value indicates whether the matrix is invertible, and thus, whether the system has a unique solution. A non-zero determinant \( (\Delta eq 0) \) means that the matrix is invertible and a unique solution exists. When the determinant is zero, the situation is different, and we have to explore other properties to determine the nature of the solutions.

Matrices

Matrices are essential in representing systems of linear equations compactly and elegantly. They consist of rows and columns, with each element representing coefficients or constant terms from the equations. In a linear system, the coefficient matrix contains the coefficients of variables, and can look like this: \[\begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix}.\]We multiply this coefficient matrix by a column matrix of variables, and set it equal to a column matrix of constants. Matrices allow the use of powerful algebraic methods to solve linear systems, such as matrix inversion or applying row reduction techniques. Understanding matrices is key to solving systems efficiently and analyzing their properties.

Unique Solutions

A system of linear equations has a unique solution if the determinant of its coefficient matrix is non-zero \( (\Delta eq 0) \). This condition ensures the matrix is invertible, meaning there is exactly one set of values for the variables that satisfy all equations simultaneously. The unique solution for such a system can be calculated using Cramer's Rule, which involves dividing determinants derived from the original matrix. This is expressed as:

• \( x_1 = \frac{\Delta_1}{\Delta} \)
• \( x_2 = \frac{\Delta_2}{\Delta} \)

where \( \Delta_1 \) and \( \Delta_2 \) are determinants formed by replacing columns in the original matrix with the constant terms. The unique intersection point of the lines represented by equations is the graphical representation of the unique solution.

Infinite Solutions

Sometimes a system may not have a unique solution but instead have infinitely many solutions. This occurs when the determinant of the coefficient matrix is zero \( (\Delta = 0) \), meaning the matrix is not invertible. For infinite solutions, the equations must be dependent, representing the same line. This means every point on one line also lies on the other. To pinpoint the condition for infinite solutions, we use the additional determinant \( \Delta_2 \). If \( \Delta_2 = 0 \), it implies the right-hand sides of the transformed equations are consistent with each other, indicating coincident lines. Thus, the system has infinitely many solutions, or in geometric terms, the lines coincide, sharing all their points. If these conditions are not met, the system might still be inconsistent, resulting in no solution. Understanding when a system leads to infinite solutions is essential in interpreting the relationships between equations.