Problem 53 - Calculus for Biology and Medicine

Question

Use the determinant to determine whether the matrix $$A=\left[\begin{array}{rr} 2 & -1 \\ -1 & 3 \end{array}\right] $$ is invertible.

Step-by-Step Solution

Answer

Matrix \( A \) is invertible because its determinant is 5, which is non-zero.

Step 1: Recall the Condition for Invertibility

A matrix is invertible if and only if its determinant is non-zero. Thus, the determinant is a key tool to determine invertibility.

Step 2: Formula for Determinant of a 2x2 Matrix

For a 2x2 matrix of the form \[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \], the determinant is given by \( \text{det}(A) = ad - bc \).

Step 3: Identify Coefficients

Identify the coefficients from the given matrix: \( a = 2 \), \( b = -1 \), \( c = -1 \), \( d = 3 \).

Step 4: Calculate the Determinant

Substitute the values into the determinant formula: \( \text{det}(A) = 2 \times 3 - (-1) \times (-1) \).

Step 5: Simplify the Expression

Calculate the determinant \( \text{det}(A) = 6 - 1 = 5 \).

Step 6: Conclusion Regarding Invertibility

Since the determinant \( \text{det}(A) = 5 \) is not zero, matrix \( A \) is invertible.

Key Concepts

Determinant

The determinant is a mathematical concept that helps us understand some important properties of matrices. It's a number (scalar) assigned to a square matrix, and it's particularly useful when studying invertibility. For a 2x2 matrix, the determinant allows us to easily decide if a matrix can be inverted or not.
If the determinant is zero, the matrix is said to be singular, meaning it does not have an inverse. Conversely, a non-zero determinant indicates that the matrix is invertible. This property makes the determinant an essential tool in linear algebra.

• An easy way to remember this: **Zero Determinant = Not Invertible** & **Non-Zero Determinant = Invertible**
• Determining the determinant is often the first step for checking matrix invertibility.

It's also worth mentioning that the determinant can tell us information about the volume that a matrix represents within a space and whether this volume can "flip" (change orientation) or not.

2x2 Matrix

Let's focus on the specific type of matrix used in this exercise: a 2x2 matrix. These matrices have two rows and two columns. They are among the simplest matrices, making them a great starting point for learning about determinants and invertibility.
The general form of a 2x2 matrix is expressed as:\[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \] Here, \( a \), \( b \), \( c \), and \( d \) represent the elements located at different positions within the matrix.

• The elements \( a \) and \( d \) represent the main diagonal.
• The term \( ad - bc \) gives the determinant of the matrix.

Understanding how to calculate properties like the determinant of these matrices is key in many applications, especially in solving linear equations where matrix mechanics are applied.

Invertible Matrix

An invertible matrix, also known as a non-singular matrix, is one that has an inverse. The inverse of a matrix is a matrix that, when multiplied with the original, results in the identity matrix. This is analogous to finding the reciprocal of a number in simple arithmetic.
For a matrix to be invertible, a crucial condition is that its determinant must be non-zero. This is because a zero determinant indicates that the matrix compresses data into a lower dimension, meaning there is no unique inverse.
Some features of invertible matrices include:

• They must be square matrices, meaning the number of rows and columns are the same.
• The product of an invertible matrix and its inverse gives the identity matrix of the same dimension.

This concept is very useful in solving systems of linear equations, transformations in geometry, and many more fields where matrix operations are key.