Problem 49 - Calculus for Biology and Medicine

Question

Find the inverse (if it exists) of $$C=\left[\begin{array}{ll} 1 & 2 \\ 3 & 6 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix \( C \) does not have an inverse because its determinant is 0.

Step 1: Verify the Determinant

To determine if the matrix \( C \) has an inverse, we first need to check its determinant. For a 2x2 matrix \( \left[ \begin{array}{cc} a & b \ c & d \end{array} \right]\), the determinant is calculated as \( ad - bc \). In this case, \( a = 1 \), \( b = 2 \), \( c = 3 \), and \( d = 6 \). Therefore, the determinant is \( (1)(6) - (2)(3) = 6 - 6 = 0 \).

Step 2: Analyze the Determinant Result

Since the determinant of matrix \( C \) is 0, the matrix is singular. A singular matrix does not have an inverse. Therefore, matrix \( C \) does not have an inverse.

Key Concepts

2x2 Matrix

A 2x2 matrix is the simplest form of a square matrix, consisting of two rows and two columns. Each element in these matrices is placed in a specific position, commonly denoted by the subscripts in terms of rows and columns, like this:

• Upper left: first row, first column (a);
• Upper right: first row, second column (b);
• Lower left: second row, first column (c);
• Lower right: second row, second column (d).

Matrices are essential in linear algebra because they represent linear transformations and systems of linear equations. In our example, the matrix \( C = \begin{bmatrix} 1 & 2 \ 3 & 6 \end{bmatrix} \) is a 2x2 matrix. Each entry indicates the coefficient of variables in linear equations or elements in a transformation matrix. Understanding how to manipulate these matrices is crucial for solving linear algebra problems.

Determinant

The determinant of a matrix is a special number calculated from its elements. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is found using the formula \( ad - bc \). In practical terms, it helps to determine if the matrix is invertible.
In our case, we calculated the determinant of the matrix \( C \), which is \( (1)(6) - (2)(3) = 0 \). The size of the determinant impacts the properties of the matrix:

• If the determinant is non-zero, the matrix is invertible.
• If the determinant is zero, the matrix is singular and not invertible.

Understanding the determinant is key because it not only indicates whether an inverse exists but also provides insight into the matrix's geometric properties, such as volume scaling factors in transformations.

Singular Matrix

A singular matrix is one that does not have an inverse. This occurs when the determinant of the matrix is zero.
For our matrix \( C \), since the determinant is \( 0 \), it indicates that the matrix is singular. Singular matrices imply that the transformation they represent cannot "undo" or reverse itself uniquely.

• In practical applications, this might indicate a loss of information or dependency between rows/columns, suggesting they are not linearly independent.
• It's important to identify singular matrices to avoid unnecessary calculations in searching for an inverse.

When solving real-world problems, if a matrix describing our system turns out to be singular, it means that the solution is either not possible or not unique, which requires considering alternative approaches like generalized inverses or pseudo-inverses.

Linear Algebra

Linear algebra is a branch of mathematics that studies vectors, vector spaces, and linear mappings between these spaces. It provides powerful tools for solving a variety of problems in both theoretical and applied settings.
The key elements in linear algebra include matrices and operations such as addition, multiplication, and finding inverses. Linear algebra is foundational for understanding systems of linear equations, which appear extensively in engineering, physics, computer science, and more.

• Matrices, the heart of linear algebra, provide a way to handle linear equations compactly and efficiently.
• Understanding inverse matrices is crucial as they allow one to solve equations of the form \( AX = B \) by transforming it to \( X = A^{-1}B \) when \( A^{-1} \) exists.

Learning linear algebra opens up opportunities to delve into more complex topics, such as eigenvalues, eigenvectors, and matrix decompositions, offering deeper insights into various scientific and engineering problems.