Problem 59 - Calculus for Biology and Medicine

Question

Write down the inverse of \(A\). $$ A=\left[\begin{array}{rr} 2 & 1 \\ -3 & -1 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of the matrix \( A \) is \( \begin{pmatrix} -1 & -1 \\ 3 & 2 \end{pmatrix} \).

Step 1: Find the determinant of A

The determinant of a 2x2 matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is calculated using the formula \( \, \text{det}(A) = ad - bc \, \). For the given matrix \( A = \begin{pmatrix} 2 & 1 \ -3 & -1 \end{pmatrix} \), the determinant is \( 2(-1) - (1)(-3) = -2 + 3 = 1 \).

Step 2: Swap and change signs

The inverse of a 2x2 matrix is found using the formula \( A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \). Replace \( a \) with \( d \) and \( d \) with \( a \), and switch the signs of \( b \) and \( c \). For our matrix, this gives us \( \begin{pmatrix} -1 & -1 \ 3 & 2 \end{pmatrix} \).

Step 3: Multiply by the reciprocal of determinant

The inverse of the matrix \( A \) is \( \frac{1}{\text{det}(A)} \) times the matrix obtained in Step 2. Since the determinant is 1, we have \( A^{-1} = 1 \times \begin{pmatrix} -1 & -1 \ 3 & 2 \end{pmatrix} = \begin{pmatrix} -1 & -1 \ 3 & 2 \end{pmatrix} \).

Key Concepts

Determinant Calculation

In linear algebra, the determinant of a matrix is a special number that can tell us important properties of the matrix, such as whether it has an inverse. For a 2x2 matrix, the determinant is calculated with a straightforward formula. If we have a matrix \( A \) as follows:

• \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \)

The determinant of \( A \), denoted \( \text{det}(A) \), is determined by multiplying the diagonal elements \( a \) and \( d \) together, and then subtracting the product of the off-diagonal elements \( b \) and \( c \). Thus, the formula is:

• \( \text{det}(A) = ad - bc \)

For the matrix given in the problem:

• \( A = \begin{pmatrix} 2 & 1 \ -3 & -1 \end{pmatrix} \)
• The determinant is calculated as: \( 2\times(-1) - 1\times(-3) = -2 + 3 = 1 \)

This positive and non-zero determinant implies that the matrix is invertible.

2x2 Matrix

A 2x2 matrix in linear algebra is a simple square matrix consisting of 2 rows and 2 columns. Understanding these matrices is crucial, as they are foundational blocks in larger calculations. They are typically written as follows:

• \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \)

Each element (\(a\), \(b\), \(c\), and \(d\)) is a component of this matrix, positioned in the specific row and column. Such matrices are widely used for

• solving systems of linear equations,
• transformations in geometry, and
• determining the invertibility of a linear system.

For a 2x2 matrix, the inverse is calculated if and only if the determinant is not zero. With the matrix given in this example, the determinant being positive allows us to find its inverse, thereby solving equations and understanding transformations better.

Linear Algebra

Linear algebra is an area of mathematics that deals with vectors, vector spaces, and linear equations. It's a critical branch of math because it lays the groundwork for applied mathematics, engineering, physics, computer science, and more.
One of the most important aspects of linear algebra is working with matrices. Matrices help solve linear systems and represent complex structures and operations. A cornerstone concept in linear algebra is matrix inversion – particularly for small matrices like 2x2 matrices.
Matrix inverses provide a way to solve matrix equations \( AX = B \) by transforming it into \( X = A^{-1}B \). For this to occur, the matrix must have a non-zero determinant. Operations such as finding an inverse help us:

• analyze geometrical transformations,
• optimize systems, and
• resolve systems of equations efficiently.

Mastering these basics in linear algebra equips students and professionals to handle more complex problems across varied scientific and technological fields.