Problem 57 - 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? $$\left[\begin{array}{ll}2 & x \\\x & x^{2}\end{array}\right]$$

Step-by-Step Solution

Answer

The matrix has no inverse when \(x = 0\).

Step 1: Understand the Inverse Condition

A matrix is invertible if its determinant is non-zero. For a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is \(ad - bc\). If this determinant is zero, the matrix has no inverse.

Step 2: Calculate the Determinant

For the matrix \(\begin{bmatrix} 2 & x \ x & x^2 \end{bmatrix}\), the determinant is computed as follows: \(2(x^2) - x(x) = 2x^2 - x^2 = x^2\).

Step 3: Determine When the Determinant is Zero

Set the determinant equal to zero to find the value(s) of \(x\) that make the matrix singular (non-invertible): \(x^2 = 0\). Solving this gives \(x = 0\).

Step 4: Compute the Inverse for Nonzero x

For \(x eq 0\), the inverse of the matrix \(\begin{bmatrix} 2 & x \ x & x^2 \end{bmatrix}\) is calculated using the formula for the inverse of a 2x2 matrix, \(A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\), which becomes: \( \frac{1}{x^2} \begin{bmatrix} x^2 & -x \ -x & 2 \end{bmatrix}\).

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated for square matrices. It provides important properties concerning the matrix, such as whether the matrix has an inverse.

For a 2x2 matrix, the determinant is found using a simple formula. Consider a matrix like this:

• Matrix: \[\begin{bmatrix} a & b \ c & d\end{bmatrix}\]

The determinant is calculated as \(ad - bc\). This number helps us determine if the matrix is invertible.

If the determinant is zero, the matrix becomes singular, which means it does not have an inverse. Understanding determinants is crucial for matrix operations, especially in linear algebra.

Invertible Matrix

An invertible matrix, also known as a non-singular matrix, is one that has an inverse. To check if a matrix is invertible, its determinant must not be zero.

The concept of an inverse matrix is like finding a reciprocal in numbers. When you multiply a matrix by its inverse, you get an identity matrix. This identity matrix acts as '1' in matrix terms.

Consider a 2x2 matrix and its inverse:

• Original Matrix: \(A\)
• Inverse Matrix: \(A^{-1}\)
• Result: \(A \times A^{-1} = I\), where \(I\) is the identity matrix.

The identity matrix for a 2x2 matrix looks like this: \[\begin{bmatrix}1 & 0 \0 & 1\end{bmatrix}\] Calculating whether a matrix has an inverse goes hand-in-hand with finding out if it's singular.

2x2 Matrix Inverse

Finding the inverse of a 2x2 matrix is a methodical process. For a matrix of form:

• \[\begin{bmatrix} a & b \c & d\end{bmatrix}\]

The inverse is defined using the determinant, provided the determinant is non-zero. The formula for the inverse is: \[A^{-1} = \frac{1}{ad - bc}\begin{bmatrix}d & -b \-c & a\end{bmatrix}\]

Let's say we have a matrix \( \begin{bmatrix}2 & x \ x & x^2 \end{bmatrix} \). The inverse can only be found if \(x^2 eq 0\), meaning \(x\) cannot be zero.

Once you calculate the inverse, it acts as a powerful tool, allowing you to solve systems of equations and more in matrix algebra.

Singular Matrix

A singular matrix is a matrix that does not have an inverse. This occurs when the determinant of the matrix is zero. Singularity implies that certain calculations cannot be done, as they require an inverse.

Mathematically, if you have a determinant equal to zero, the matrix cannot be reversed or "undone" in linear algebra operations.

Consider the 2x2 example matrix:

• \[\begin{bmatrix} 2 & x \ x & x^2 \end{bmatrix}\]

If we set up the determinant and get \(x^2 = 0\), then \(x = 0\) renders the matrix singular.

To avoid working with singular matrices, it's essential to ensure the determinant is not zero when performing matrix operations. Understanding singular matrices is crucial as it helps avoid potential pitfalls in matrix computations.