Problem 41 - Algebra and Trigonometry

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 if \(x = 0\). Otherwise, the inverse is \(\left[\begin{array}{cc}1 & -\frac{1}{x} \\ -\frac{1}{x} & \frac{2}{x^2} \end{array}\right]\)."

Step 1: Understand the Matrix Inversion

For a 2x2 matrix \(A = \left[\begin{array}{cc}a & b \ c & d \end{array}\right]\), the inverse exists if its determinant is non-zero. The formula for the inverse, when it exists, is \(A^{-1} = \frac{1}{ad-bc}\left[\begin{array}{cc}d & -b \ -c & a \end{array}\right]\). The determinant of this matrix is given by \(ad - bc\).

Step 2: Apply to the Given Matrix

Given the matrix \(A = \left[\begin{array}{cc}2 & x \ x & x^2 \end{array}\right]\), we identify \(a = 2\), \(b = x\), \(c = x\), and \(d = x^2\). The determinant of this matrix is \(2x^2 - x \times x = 2x^2 - x^2 = x^2\).

Step 3: Determine When the Inverse Exists

The matrix will have an inverse only if the determinant is non-zero. Set \(x^2 eq 0\). This means \(x eq 0\).

Step 4: Find the Inverse Matrix when it Exists

If \(x eq 0\), the inverse of \(A\) is calculated as:\[ A^{-1} = \frac{1}{x^2} \left[\begin{array}{cc}x^2 & -x \ -x & 2 \end{array}\right] = \left[\begin{array}{cc}1 & -\frac{x}{x^2} \ -\frac{x}{x^2} & \frac{2}{x^2} \end{array}\right] = \left[\begin{array}{cc}1 & -\frac{1}{x} \ -\frac{1}{x} & \frac{2}{x^2} \end{array}\right]\]

Key Concepts

2x2 matrix

A 2x2 matrix is a type of matrix that has two rows and two columns. It is typically denoted by the arrangement \(\left[\begin{array}{cc}a & b \ c & d \end{array}\right]\). Here, \(a\), \(b\), \(c\), and \(d\) are elements of the matrix and can be any numbers or parameters. Understanding the structure of a 2x2 matrix is crucial because it serves as a building block of more complicated matrices.

• The element at the top left is \(a\),
• The top right is \(b\),
• The bottom left is \(c\), and
• The bottom right is \(d\).

In mathematical operations, 2x2 matrices are key for various functions such as determining matrix inverse or calculating determinants. These operations play important roles in solving systems of linear equations and in transformations used in computer graphics.

determinant

The determinant of a 2x2 matrix is a special number calculated from its elements, defined as \(ad - bc\). It provides crucial information about the matrix.To compute the determinant for a matrix \(\left[\begin{array}{cc}a & b \ c & d \end{array}\right]\), you use:\[\text{Det}(A) = ad - bc\]In our specific problem, the determinant helps us determine whether the given matrix has an inverse.Key properties of determinants:

• If the determinant is non-zero, the matrix has an inverse.
• If the determinant is zero, the matrix does not have an inverse (it is singular).

In practice, determinants are invaluable for understanding matrix behavior in geometric transformations, and they help define matrix invertibility.

inverse matrix

An inverse matrix is a matrix that, when multiplied by the original matrix, results in an identity matrix. For a 2x2 matrix \(A\), its inverse is represented by the formula:\[A^{-1} = \frac{1}{ad-bc}\left[\begin{array}{cc}d & -b \ -c & a \end{array}\right]\]Here, \(ad-bc\) is the determinant, and it's crucial that this value is non-zero for the inverse to exist.Steps to find the inverse:

• Calculate the determinant \(ad-bc\).
• Ensure the determinant is not zero.
• Apply the formula for the inverse.

The inverse matrix is largely used in solving systems of linear equations, particularly when dealing with equations that are interrelated, allowing one to find solutions effectively.

non-zero determinant

A non-zero determinant implies that a matrix is invertible, meaning it has an inverse. This concept is crucial in linear algebra and ensures that the transformations represented by the matrix are reversible.For any matrix, particularly our 2x2 matrix \(\left[\begin{array}{cc}a & b \ c & d \end{array}\right]\), the determinant must be non-zero (\(ad-bc eq 0\)) for the matrix to be defined as invertible. Importance of a non-zero determinant:

• It indicates that the matrix has full rank and maps space in a non-degenerate way.
• An invertible matrix implies that you can solve the system of equations uniquely.

In practical terms, working with non-zero determinants helps verify the existence of solutions and manage matrix operations in computational tasks.