Problem 70 - Calculus for Biology and Medicine

Question

Find the inverse matrix to each given matrix if the inverse matrix exists. $$ A=\left[\begin{array}{rrr} -1 & 0 & 2 \\ -1 & -2 & 3 \\ 0 & 2 & -1 \end{array}\right] $$

Step-by-Step Solution

Answer

Matrix \( A \) is not invertible because its determinant is 0.

Step 1: Identify the biological concept

Determine which biology area this addresses.

Step 2: Recall relevant principles

Review key concepts.

Step 3: Apply knowledge

Use biological reasoning.

Step 4: State the answer

The answer is: Matrix \( A \) is not invertible because its determinant is 0.

Key Concepts

Determinant Calculation

The determinant of a matrix is a special number that is calculated from its elements. It is crucial in determining whether a square matrix is invertible. To find the determinant of a 3x3 matrix, like the given matrix \( A \), we use a method called expansion by minors. This involves breaking down the matrix into smaller 2x2 matrices, known as minors, by eliminating one row and one column at a time.

For example, in the provided exercise, matrix \( A \) is given by:
\[ A = \begin{bmatrix} -1 & 0 & 2 \ -1 & -2 & 3 \ 0 & 2 & -1 \end{bmatrix} \]
To compute its determinant, \( \text{det}(A) \), we take each element of the first row, multiply it by the determinant of its corresponding minor, and alternate signs as follows:

• The first minor, formed from the first row and column removed, is: \( \begin{vmatrix} -2 & 3 \ 2 & -1 \end{vmatrix} \).
• The second element, 0, multiplies its own minor's determinant.
• The third minor, comes from removing its row and column, is: \( \begin{vmatrix} -1 & -2 \ 0 & 2 \end{vmatrix} \).

These calculations ultimately lead to the overall determinant value, confirming if a matrix can be inverted.

Invertible Matrix

An invertible matrix is one that has an inverse. Not all matrices are invertible, and the first step to check invertibility is to calculate the matrix's determinant.

For a matrix to be invertible:

• Its determinant must not be zero.
• A non-zero determinant indicates that the matrix is full-rank, meaning its rows are linearly independent.

If the determinant is zero, like in the exercise with matrix \( A \), no inverse exists because the matrix is singular, meaning some rows or columns are dependent on each other. This renders operations like division impossible, as they would be in real number scenarios with dividing by zero.

In scenarios where the determinant is non-zero, matrix inversion can proceed using various methods like the adjugate method or Gaussian elimination, providing a matrix that, when multiplied with the original, results in the identity matrix.

Linear Algebra

Linear algebra is a branch of mathematics concerning vectors, vector spaces, and linear transformations between them. This field studies matrices and their properties, such as determinants and inverses.

Central to linear algebra:

• Matrices are used to represent and solve systems of linear equations.
• Operations like finding determinants, checking invertibility, and computing inverses are fundamental.

These concepts are crucial in applications ranging from engineering to computer science.

The ability to determine if a matrix is invertible has practical applications such as modeling physical systems, solving equilibrium equations, and many more. This is why exercises like calculating determinants and checking invertibility, like the one provided, are fundamental in linear algebra education. Understanding these concepts deeply enhances problem-solving abilities and supports higher-level topics within mathematics.