Problem 60 - Precalculus: Mathematics for Calculus

Question

When Do Matrices Have Inverses? Find the inverse of the matrix. For what value(s) of \(x,\) if any, does the matrix have no inverse? $$\left[\begin{array}{cc}x & 1 \\\\-x & \frac{1}{x-1}\end{array}\right]$$

Step-by-Step Solution

Answer

The matrix has no inverse for \(x = 0\) and \(x = 1\).

Step 1: Determine Matrix Inverse Existence

A matrix has an inverse if and only if its determinant is not zero. Let's find the determinant of the given matrix:\[\text{Matrix} = \begin{bmatrix} x & 1 \ -x & \frac{1}{x-1} \end{bmatrix}\]The determinant \( \text{det}(A) \) is calculated as:\[\text{det}(A) = (x)\left(\frac{1}{x-1}\right) - (1)(-x) = \frac{x}{x-1} + x = \frac{x + x(x-1)}{x-1} = \frac{x^2}{x-1}\]

Step 2: Identify Values Making Determinant Zero

To find values of \(x\) that make the determinant zero, set the determinant equal to zero and solve:\[\frac{x^2}{x-1} = 0 \]A fraction is zero only if its numerator is zero, therefore, solve:\[x^2 = 0\]\[x = 0\]

Step 3: Consider Denominator Restrictions

Additionally, we must ensure that the determinant expression is defined, meaning the denominator cannot be zero:\[x-1 eq 0 \\Rightarrow x eq 1\]

Step 4: Conclusion on Inverse Existence

From Step 2 and Step 3, we conclude:- The matrix has no inverse for \( x = 0 \), due to making the determinant zero.- The determinant is undefined (and thus no inverse exists) for \( x = 1 \) because the denominator is zero.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements. It plays a crucial role in various matrix operations, especially when identifying whether a matrix has an inverse. For a 2x2 matrix, finding the determinant is straightforward:

Given matrix:\[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]The determinant, denoted as \( \text{det}(A) \), is calculated by \( ad - bc \).

For a matrix to have an inverse, its determinant must be non-zero. If the determinant is zero, the matrix is said to be singular, meaning it does not have an inverse. When working with matrices, always check the determinant to determine the matrix's invertibility.

If you're dealing with a matrix that contains variables, like the example given, you'll first need to compute the determinant and then solve any resulting equations to find out for which values of the variable the determinant equals zero. This will help in identifying when the matrix is singular, or in simpler terms, when it doesn't have an inverse.

Matrix Algebra

Matrix algebra is a branch of mathematics that extends algebraic theories to matrices. It involves operations like addition, subtraction, and multiplication of matrices, as well as more complex concepts such as determinants and inverses.

Mathematical operations involving matrices follow specific rules, making it crucial to understand these guidelines for solving problems involving matrices:

• Addition and Subtraction: Only matrices of the same dimension can be added or subtracted by adding or subtracting corresponding elements.


• Multiplication: The multiplication of matrices requires that the number of columns in the first matrix matches the number of rows in the second matrix. The resulting product matrix will have dimensions corresponding to the number of rows of the first matrix and the number of columns of the second matrix.


• Scalar Multiplication: Every element of the matrix is multiplied by the scalar.

Understanding these basics makes it easier to manipulate matrices to solve systems of equations or compute matrix inverses, among many other applications in mathematics and applied sciences.

Inverse of a Matrix

The inverse of a matrix, denoted as \( A^{-1} \), is a matrix that, when multiplied with the original matrix \( A \), yields the identity matrix. Not every matrix has an inverse, and determining the existence of an inverse is tied directly to the determinant.

For a 2x2 matrix \( A \):\[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]The inverse \( A^{-1} \) exists if and only if the determinant \( \text{det}(A) \) is non-zero and is given by:\[ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \]

The identity matrix for a 2x2 matrix is:\[ \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]

This property means that for each non-zero determinant matrix, a unique inverse exists, reversing the effects of the initial matrix transformation. It’s crucial to remember that matrices must be square (same number of rows and columns) to have inverses. When a matrix doesn’t meet these criteria or its determinant is zero, it simply doesn’t have an inverse. Identifying this early helps avoid errors in matrix calculations.