Problem 39 - Algebra and Trigonometry

Question

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

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: Determine the formula for the inverse of a 2x2 matrix

For a 2x2 matrix of the form \( \begin{bmatrix} e & f \ g & h \end{bmatrix} \), its inverse is found using the formula:\[\text{Inverse} = \frac{1}{\text{det}} \begin{bmatrix} h & -f \ -g & e \end{bmatrix}\]where \( \text{det} = eh - fg \) is the determinant of the matrix.

Step 2: Calculate the determinant of the given matrix

The given matrix is \( \begin{bmatrix} a & -a \ a & a \end{bmatrix} \). Calculate the determinant:\[\text{det} = (a)(a) - (a)(-a) = a^2 + a^2 = 2a^2\]

Step 3: Check if the determinant is non-zero

The determinant \( 2a^2 eq 0 \) since \( a eq 0 \). Hence, the matrix is invertible.

Step 4: Apply the inverse formula

Substitute the elements of the matrix into the inverse formula:\[\text{Inverse} = \frac{1}{2a^2} \begin{bmatrix} a & a \ -a & a \end{bmatrix}\]

Step 5: Simplify the inverse matrix

Multiply each element by \( \frac{1}{2a^2} \):\[\text{Inverse} = \begin{bmatrix} \frac{a}{2a^2} & \frac{a}{2a^2} \ \frac{-a}{2a^2} & \frac{a}{2a^2} \end{bmatrix} = \begin{bmatrix} \frac{1}{2a} & \frac{1}{2a} \ \frac{-1}{2a} & \frac{1}{2a} \end{bmatrix}\]

Step 6: Present the final inverse matrix

The inverse of the matrix \( \begin{bmatrix} a & -a \ a & a \end{bmatrix} \) is:\[\begin{bmatrix} \frac{1}{2a} & \frac{1}{2a} \ \frac{-1}{2a} & \frac{1}{2a} \end{bmatrix}\]

Key Concepts

2x2 Matrix

A 2x2 matrix is a simple form of a matrix representing a rectangular array of numbers. It has two rows and two columns, making it highly manageable for various computations. In general, a 2x2 matrix might look like this:

• First row: contains two elements, often denoted as "e" and "f".
• Second row: contains two elements as well, usually denoted as "g" and "h".

Such a matrix can be written as:\[\begin{bmatrix} e & f \ g & h \end{bmatrix}\]This small-sized matrix is particularly interesting because its elements interact to determine important properties like the determinant and inverse.
The elements in the matrix aren’t just arbitrary; they can represent various quantities, such as coefficients in a system of equations.
Understanding a 2x2 matrix is a crucial foundation when learning about matrices in general.

Matrix Inversion

Matrix inversion is the process of finding another matrix that, when multiplied by the original matrix, results in the identity matrix.
For a 2x2 matrix, there’s a specific formula to find its inverse, providing that the determinant is not zero. Here’s what to do:

• First, determine the determinant of the matrix. If the determinant is zero, the matrix is not invertible.
• Then, rearrange the elements according to the formula for the inverse.
• Finally, multiply the rearranged matrix by the reciprocal of the determinant.

To put it simply, the inverse of a 2x2 matrix \( \begin{bmatrix} e & f \ g & h \end{bmatrix} \) is computed as:\[\text{Inverse} = \frac{1}{\text{det}} \begin{bmatrix} h & -f \ -g & e \end{bmatrix}\]Ensuring that a matrix is invertible before attempting to find the inverse is crucial.
This fundamental concept is widely used in solving systems of linear equations and various applications in engineering and computer science.

Determinant of a Matrix

The determinant of a 2x2 matrix is a special number that provides essential information about the matrix. For a matrix:\[\begin{bmatrix} e & f \ g & h \end{bmatrix}\]The determinant is calculated using the formula:\[\text{det} = eh - fg\]This scalar value plays a crucial role in our ability to find the inverse of a matrix. Negative, positive, or zero values of the determinant can affect the properties of the matrix.

• If the determinant is zero, the matrix is singular and does not have an inverse.
• Non-zero determinants indicate that the matrix is nonsingular, meaning it can be inverted.

In practical terms, the determinant can be seen as a measure for understanding transformations represented by the matrix, such as scaling in 2D space.
Deep understanding of the determinant is essential for working with transformations and systems of linear equations.