Problem 41 - Precalculus: Mathematical 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 for \( x = 0 \). For other values of \( x \), the inverse exists.

Step 1: Understand the Condition for Inverse

A matrix has an inverse if and only if its determinant is non-zero. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is given by \( ad - bc \).

Step 2: Write the Expression for the Determinant

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

Step 3: Solve for Zero Determinant

To find the values of \( x \) for which the matrix has no inverse, set the determinant \( x^2 = 0 \). This gives \( x = 0 \).

Step 4: Conclusion on Inverse

The matrix does not have an inverse when \( x = 0 \). For other values of \( x \), the matrix is invertible, and the inverse can be calculated using the formula for a 2x2 inverse: \[A^{-1} = \frac{1}{x^2} \begin{bmatrix} x^2 & -x \ -x & 2 \end{bmatrix}\]

Key Concepts

Determinant

The determinant is a special number that can be calculated from a square matrix. It provides insights into properties like invertibility of the matrix. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated with the formula \( ad - bc \). This result is crucial because if the determinant is zero, the matrix does not have an inverse. A non-zero determinant indicates that the matrix can be inverted.

• The determinant reflects how much the area or volume is scaled when you apply the matrix transformation.
• For square matrices larger than 2x2, the process involves more complex operations, but the core principle remains the same.

In the context of our specific exercise, the determinant of the matrix \( \begin{bmatrix} 2 & x \ x & x^2 \end{bmatrix} \) is found by substituting into the formula: \( 2 \cdot x^2 - x \cdot x = x^2 \). Solving for when this equals zero allows us to find conditions under which the matrix has no inverse, namely when \( x = 0 \).

2x2 Matrix

A 2x2 matrix is a simple square matrix with two rows and two columns. It's often used to represent transformations or operations in two-dimensional space. Each component of a 2x2 matrix can significantly impact its behavior, such as its ability to be inverted or its effect on vectors.

• When determining properties like invertibility, concise formulas can be employed, such as for the determinant or the inverse.
• 2x2 matrices are foundational in linear algebra and often serve as the starting point for understanding more complex systems.

For example, the matrix provided in our exercise, \( \begin{bmatrix} 2 & x \ x & x^2 \end{bmatrix} \), changes based on the value of \( x \). This alters both its determinant and overall transformation properties, illustrating the simple but potent nature of 2x2 matrices.

Matrix Invertibility

Matrix invertibility is a vital concept in linear algebra. It essentially asks whether you can "undo" the transformation represented by a matrix. A matrix is invertible if there is another matrix that when multiplied yields the identity matrix. For 2x2 matrices, a quick determinant check tells us about invertibility: if the determinant is zero, the matrix is not invertible.

• An invertible matrix allows systems of linear equations to be solved uniquely.
• Inversion is equivalent to reversing some operation or transformation the matrix performs.

In our exercise, the matrix \( \begin{bmatrix} 2 & x \ x & x^2 \end{bmatrix} \) is invertible whenever \( x eq 0 \). This is because the determinant \( x^2 \) must not equal zero. If \( x = 0 \), the entry would result in a zero determinant, prohibiting inversion, thus no unique solution for the system it represents can be assured.