Problem 45 - Calculus for Biology and Medicine

Question

Let $$\boldsymbol{A}=\left[\begin{array}{rr} -\mathbf{1} & \mathbf{1} \\ \mathbf{2} & \mathbf{3} \end{array}\right], \quad \boldsymbol{B}=\left[\begin{array}{ll} \mathbf{2} & \mathbf{0} \\ \mathbf{3} & \mathbf{2} \end{array}\right]$$ Find the inverse (if it exists) of \(A\).

Step-by-Step Solution

Answer

The inverse of \( A \) is \( \begin{bmatrix} -\frac{3}{5} & \frac{1}{5} \\ \frac{2}{5} & -\frac{1}{5} \end{bmatrix}. \)

Step 1: Determinant of A

To find the inverse of matrix \( A \), we first need to compute its determinant. The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by \( ad - bc \). For matrix \( A \), we have: \[-1(3) - (1)(2) = -3 - 2 = -5.\] Since the determinant is not zero (\(-5eq 0\)), \( A \) is invertible.

Step 2: Formula for the Inverse of A

The inverse of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by \( \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). Now that we have the determinant from the previous step, we can plug in the values from matrix \( A \).

Step 3: Compute the Inverse Matrix

Using the formula from Step 2 and the determinant \(-5\), the inverse of \( A \) is calculated as: \[ \frac{1}{-5} \begin{bmatrix} 3 & -1 \ -2 & -1 \end{bmatrix} \] \[ = \begin{bmatrix} -\frac{3}{5} & \frac{1}{5} \ \frac{2}{5} & -\frac{1}{5} \end{bmatrix}. \] This matrix is the inverse of \( A \).

Key Concepts

Matrix Inverse

When we talk about a matrix inverse, we're looking for a matrix that, when multiplied by the original matrix, results in an identity matrix. An identity matrix is a special kind of matrix that acts as the multiplicative identity in matrix algebra, much like the number 1 does in regular arithmetic. For a 2x2 matrix, the inverse is computed using a special formula. This formula requires you to first find the determinant, which we'll talk about shortly.

The process of finding a matrix inverse only works for square matrices where the determinant is non-zero. Hence, calculating the determinant comes first. If the determinant is zero, the matrix is said to be singular, which means it does not have an inverse.

• Speaks to the "undoing" or reversal of a linear transformation.
• Applicable only when the determinant is not zero.
• The formula to calculate the inverse involves the elements of the original matrix and its determinant.

Understanding the inverse is crucial in solving matrix equations, such as those commonly encountered in linear algebra and engineering problems.

Determinant Calculation

The determinant of a matrix is a special number that can tell us a lot about the matrix itself. For a 2x2 matrix, the determinant is calculated using the formula:\[\text{det} \begin{bmatrix} a & b \ c & d \end{bmatrix} = ad - bc\]This formula gives a single numerical value. In our case, for matrix \( A = \begin{bmatrix} -1 & 1 \ 2 & 3 \end{bmatrix} \), the determinant is:
\[(-1)(3) - (1)(2) = -3 - 2 = -5\]This shows that the determinant is \(-5\).

• A non-zero determinant indicates that the matrix is invertible.
• In simpler terms, this means the matrix can be "reversed" or "undone" via multiplication by its inverse.
• The determinant is pivotal in determining the properties and behaviors of a matrix, affecting concepts like linear dependency and transformation capabilities.

The determinant is an essential figure when considering linear transformations and provides insights into the scale change of transformations applied by matrices.

2x2 Matrices

2x2 matrices are among the simplest forms of matrices, yet they hold foundational importance in linear algebra. Represented as a grid containing 2 rows and 2 columns, they can model a lot of basic transformations.

2x2 matrices are crucial because they serve as the building blocks for higher-dimensional matrix operations. They also make mathematical operations, such as finding inverses and determinants, more manageable given their reduced size.

• 2x2 matrices can represent basic operations like rotations and translations in two-dimensional space.
• Despite their simplicity, they introduce fundamental concepts used in more complex matrix manipulation.
• Understanding 2x2 matrices aids in grasping more advanced matrix operations and applications seen in various scientific fields, including computer graphics and theoretical physics.

While simple, the 2x2 matrices offer an entry point into understanding the much larger and more complex world of matrix algebra, serving both practical and theoretical purposes in mathematics.