Problem 65 - Calculus for Biology and Medicine

Question

Use the determinant to determine whether $$ C=\left[\begin{array}{ll} 1 & 3 \\ 1 & 3 \end{array}\right] $$ is invertible. If it is invertible, compute its inverse. In either case, solve \(C X=\mathbf{0}\).

Step-by-Step Solution

Answer

The matrix is not invertible; solutions are \(X = \begin{bmatrix} -3t \\ t \end{bmatrix}\).

Step 1: Identify the Matrix

The matrix given is \(C = \begin{bmatrix} 1 & 3 \ 1 & 3 \end{bmatrix}\). This is a 2x2 matrix.

Step 2: Compute the Determinant

The formula for the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is \( ad - bc \). For our matrix: \[det(C) = 1 \times 3 - 1 \times 3 = 3 - 3 = 0\] Since the determinant is 0, the matrix is not invertible.

Step 3: Solve the Equation \(CX = \mathbf{0}\)

Since the matrix is not invertible and its determinant is zero, it implies that there are infinitely many solutions to the equation \(CX = \mathbf{0}\). Let's set up the equation:\[ \begin{bmatrix} 1 & 3 \ 1 & 3 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix}\]This results in the following system of linear equations:1. \(1x_1 + 3x_2 = 0\)2. \(1x_1 + 3x_2 = 0\)Since both equations are the same, we have the freedom to choose a value for \(x_2\), e.g., let \(x_2 = t\). Substitute this into the equation:\[x_1 + 3t = 0 \ x_1 = -3t\]Thus, the solution in terms of a parameter \(t\) is:\( X = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} -3t \ t \end{bmatrix}\)where \(t \in \mathbb{R}\).

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that gives important information about the matrix. Specifically, it tells you whether a matrix is invertible or not. If the determinant is zero, the matrix does not have an inverse. This is crucial when trying to solve systems of equations or dealing with matrix multiplications in more advanced algebra.
For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated using the formula \( ad - bc \). This simple calculation can have large implications on the properties of the matrix. For example, in the matrix \( C = \begin{bmatrix} 1 & 3 \ 1 & 3 \end{bmatrix} \), computing the determinant gives \( 1 \times 3 - 1 \times 3 = 0 \). The result of 0 means that the matrix \( C \) is not invertible.

Systems of Linear Equations

Matrices are a powerful tool for solving systems of linear equations. When you have a system where each equation has the same number of unknowns as there are equations, it can often be written in matrix form as \( AX = B \).
In the specific case where \( A \) is not invertible, like the matrix \( C \) with a determinant of 0, the system might have special kinds of solutions.
With the matrix \( C = \begin{bmatrix} 1 & 3 \ 1 & 3 \end{bmatrix} \) from the exercise, the system \( CX = \mathbf{0} \) has infinitely many solutions. This means rather than pinpointing one exact solution, the system gives us a whole family of solutions bounded by a parameter.

• If two equations yield the same result as in this example, we can solve for one of the variables in terms of the other.
• We introduce a parameter \( t \) such as setting \( x_2 = t \), then solving for \( x_1 \) gives \( x_1 = -3t \).

Thus, we express the solution in terms of \( t \), showing an infinite number of solutions for every choice of \( t \).

2x2 Matrix

A 2x2 matrix is a simple yet highly valuable construct in mathematics, particularly in linear algebra. It has two rows and two columns, making it one of the easiest types of matrices to work with. Despite the simplicity, understanding it well is key because it lays the groundwork for tackling larger matrices.
Each entry in the matrix can represent constants from equations, coefficients, or even transformations in geometry. The matrix \( C = \begin{bmatrix} 1 & 3 \ 1 & 3 \end{bmatrix} \) is a 2x2 matrix where each row or column corresponds to coefficients from a set of linear equations.

• The calculation of the determinant, as previously explained, is straightforward with a 2x2 matrix and directly influences whether the matrix is invertible.
• Understanding a 2x2 matrix equips us with the skills needed to solve simple systems of linear equations or transformations.

This matrix, although small, is an essential piece of the puzzle in linear algebra, helping to solve or simplify broader mathematical problems.