Problem 41 - Algebra and Trigonometry

Question

Find the inverse of the matrix. $$ \begin{array}{l}{\left[\begin{array}{rr}{a} & {-a} \\ {a} & {a}\end{array}\right]} \\ {(a \neq 0)}\end{array} $$

Step-by-Step Solution

Answer

The inverse is \( \begin{bmatrix} \frac{1}{2a} & \frac{1}{2a} \\ -\frac{1}{2a} & \frac{1}{2a} \end{bmatrix} \).

Step 1: Determining the Matrix Inverse Formula

The inverse of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by \( \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). We assume the determinant \( ad-bc eq 0 \) is non-zero.

Step 2: Identify \( a, b, c, d \) from Matrix

For the given matrix \( \begin{bmatrix} a & -a \ a & a \end{bmatrix} \), identify the elements: \( a = a, b = -a, c = a, d = a \).

Step 3: Calculate the Determinant

Calculate the determinant: \( ad - bc = a \cdot a - (-a) \cdot a = a^2 + a^2 = 2a^2 \). Since \( a eq 0 \), the determinant \( 2a^2 eq 0 \), so the inverse exists.

Step 4: Apply Inverse Formula

Using the formula from Step 1, the inverse is given by \[ \frac{1}{2a^2} \begin{bmatrix} a & a \ -a & a \end{bmatrix} \].

Step 5: Simplify the Inverse Matrix

Simplify the inverse matrix by dividing each element by \(2a^2\):\[ \begin{bmatrix} \frac{a}{2a^2} & \frac{a}{2a^2} \ \frac{-a}{2a^2} & \frac{a}{2a^2} \end{bmatrix} = \begin{bmatrix} \frac{1}{2a} & \frac{1}{2a} \ -\frac{1}{2a} & \frac{1}{2a} \end{bmatrix} \].

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements. For a 2x2 matrix, the determinant plays a crucial role in matrix algebra, especially when finding the inverse of the matrix. Knowing how to compute it is essential for determining if a matrix is invertible or not.
The formula for the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is \( ad - bc \). This formula uses multiplication and subtraction of the specific elements from the matrix.

• If the determinant is zero, the matrix does not have an inverse. We call such matrices "singular."
• If it's non-zero, the matrix does have an inverse, indicating it's "non-singular."

In the example provided, the determinant calculation involved the elements \( a = a \), \( b = -a \), \( c = a \), and \( d = a \). Substituting these into the determinant formula, \( ad - bc = a^2 + a^2 = 2a^2 \), shows it is non-zero (as long as \( a eq 0 \)). Thus, the matrix can be inverted.

2x2 Matrix

A 2x2 matrix is a simple way to represent a set of numbers, organized in two rows and two columns. It is one of the basic forms of a matrix used in many fields such as mathematics, computer science, and engineering.
A 2x2 matrix can represent linear transformations and is the simplest type of matrix that can have an inverse.

• It has four elements, which are typically designated as \( a \), \( b \), \( c \), and \( d \) in mathematical notation: \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \).
• Understanding the structure and elements of the matrix is crucial for performing operations like addition, subtraction, multiplication, and finding the inverse.

In the exercise, the given matrix \( \begin{bmatrix} a & -a \ a & a \end{bmatrix} \) is a typical example of a 2x2 matrix. With each element clearly defined, you can then proceed to calculate necessary properties like the determinant and inverse.

Matrix Algebra

Matrix algebra is the branch of mathematics that deals with matrix operations. It provides the fundamental operations needed to manipulate matrices, which are essential in solving mathematical problems.
Key operations include:

• **Addition and Subtraction:** Matrices of identical sizes can be added or subtracted element-wise.
• **Multiplication:** Matrices can be multiplied by another matrix or by a scalar, although specific rules apply.
• **Inversion:** This involves finding another matrix that, when multiplied with the original, results in the identity matrix. It is possible only for "non-singular" matrices.

Inverse matrix calculation is a pivotal application of matrix algebra. It requires special attention to the determinant, as shown in the step-by-step solution. The existence of the inverse depends precisely on the non-zero determinant. Matrix algebra allows us to solve linear equations, perform transformations, and analyze systems in multiple dimensions.