Problem 43 - Precalculus Mathematics for Calculus

Question

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 Matrix

We have a 2x2 matrix \( A \) given by: \[A = \begin{bmatrix} 2 & x \ x & x^2 \end{bmatrix}\] where \( x \) is a variable. Our task is to find its inverse.

Step 2: Condition for Inverse of a Matrix

A 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) has an inverse if and only if its determinant is non-zero. If the determinant is zero, the matrix does not have an inverse.

Step 3: Calculate the Determinant of Matrix

The determinant of a matrix \( A \) is calculated as:\[\text{det}(A) = ad - bc\]For our matrix, \( a = 2 \), \( b = x \), \( c = x \), \( d = x^2 \). Thus,\[\text{det}(A) = (2)(x^2) - (x)(x) = 2x^2 - x^2 = x^2\]

Step 4: Determine when the Matrix has no Inverse

The matrix has no inverse when \( \text{det}(A) = 0 \). Hence, we solve:\[x^2 = 0\]This gives us \( x = 0 \).

Step 5: Conclusion on Values of \( x \)

For the value \( x = 0 \), the determinant is zero, so the matrix has no inverse. For any \( x eq 0 \), the determinant is non-zero, and the matrix has an inverse.

Key Concepts

2x2 Matrix

A 2x2 matrix is one of the simplest forms of matrices we deal with in matrix algebra. It consists of two rows and two columns, forming a square. Each element in the matrix can be a number or a variable. The general form of a 2x2 matrix is:\[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]

• First row: Contains the elements \(a\) and \(b\).
• Second row: Contains the elements \(c\) and \(d\).

Analyzing a 2x2 matrix is fundamentally about understanding how the arrangement and values of these numbers or variables impact overall properties like determinants and inverses.
In this context, it is important to engage with each of these matrices fully, recognizing how they apply to various algebraic calculations, such as determining an inverse.

Determinant

The determinant of a matrix is a special number that can be calculated from its elements. For a 2x2 matrix like \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant, denoted as \( \text{det}(A) \), is calculated with a simple formula:

• \( \text{det}(A) = ad - bc \)

This means you multiply the diagonals and then subtract the results:

• First diagonal product: \( a \times d \)
• Second diagonal product: \( b \times c \)

For the matrix to have an inverse, its determinant must not be zero. When the determinant equals zero, the matrix is said to be "singular" and lacks an inverse.
In practice, the determinant often serves as a prerequisite check before proceeding with further calculations in matrix algebra, such as finding an inverse.

Matrix Algebra

Matrix algebra involves various operations and principles to manage matrices. Key operations include addition, subtraction, multiplication, and finding inverses.
Here's a closer look at how these operations manifest, especially concerning inverses:

• A matrix has an inverse only if it is square (same number of rows and columns).
• The determinant must not be zero for an inverse to exist.
• The process of finding an inverse applies specific formulas suited for the matrix size.

For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the inverse \( A^{-1} \) is found using the formula:

• \( A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \), where \( ad-bc \) is the determinant.

Understanding these foundations in matrix algebra equips you to solve a variety of problems involving matrices, enhancing your ability to navigate more complex mathematical terrains.