Problem 52 - Precalculus

Question

find \(A^{-1},\) if possible. $$A=\left[\begin{array}{ll}3 & 1 \\\2 & 1\end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of matrix \( A \) is \( A^{-1} = \begin{bmatrix} 1 & -1 \\ -2 & 3 \end{bmatrix}.\)

Step 1: Check if the Matrix is Invertible

To find the inverse of matrix \( A \), we must first determine if it is invertible. A matrix is invertible if its determinant is non-zero. Calculate the determinant of matrix \( A \):\[\text{det}(A) = (3)(1) - (1)(2) = 3 - 2 = 1.\]Since \( \text{det}(A) = 1 eq 0 \), the matrix is invertible.

Step 2: Set Up the Formula for the Inverse

The inverse of a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by\[A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}.\]For our matrix \( A = \begin{bmatrix} 3 & 1 \ 2 & 1 \end{bmatrix} \), identify \( a = 3, b = 1, c = 2, \) and \( d = 1 \).

Step 3: Calculate the Inverse Matrix

Using the formula for the inverse, plug in the identified values. We have already found \( \text{det}(A) = 1 \), so\[A^{-1} = \frac{1}{1} \begin{bmatrix} 1 & -1 \ -2 & 3 \end{bmatrix} = \begin{bmatrix} 1 & -1 \ -2 & 3 \end{bmatrix}.\]

Step 4: Verify the Result

To ensure the calculated inverse is correct, verify that \( A \cdot A^{-1} = I \), where \( I \) is the identity matrix:\[\begin{bmatrix} 3 & 1 \ 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & -1 \ -2 & 3 \end{bmatrix} = \begin{bmatrix} (3)(1) + (1)(-2) & (3)(-1) + (1)(3) \ (2)(1) + (1)(-2) & (2)(-1) + (1)(3) \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} = I.\]Since the product is the identity matrix, our inverse is correct.

Key Concepts

Determinant

The determinant is a special number assigned to a square matrix. It's essential because it helps determine if a matrix is invertible or not.
In the context of a 2x2 matrix, the determinant can be calculated using the formula: for \[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \]\ the determinant \[ \text{det}(A) = ad - bc. \]This calculation gives us a single number.

• If the determinant is zero, the matrix is not invertible, meaning it doesn't have an inverse.
• If the determinant is not zero, the matrix is invertible.

For instance, for the matrix \[ A = \begin{bmatrix} 3 & 1 \ 2 & 1 \end{bmatrix}, \]the determinant is calculated as follows: \[ \text{det}(A) = (3 \times 1) - (1 \times 2) = 3 - 2 = 1. \]Since this is not zero, the matrix is invertible.

Invertible Matrices

A matrix is said to be invertible if it has an inverse. Not all matrices have inverses.
An invertible matrix, when multiplied by its inverse, will produce the identity matrix. This property is symbolized mathematically as \( A \cdot A^{-1} = I, \) where \( I \) is the identity matrix.

• For a matrix to be invertible, its determinant must be non-zero.
• The concept of an invertible matrix is crucial in solving linear equations and transforming geometric spaces.

The inverse of a 2x2 matrix \[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \]can be found using the formula: \[ A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}. \] This inverse will only exist if \( \text{det}(A) eq 0. \)

Identity Matrix

The identity matrix is a fundamental concept in linear algebra. It acts like the number 1 in arithmetic.
When you multiply any matrix by the identity matrix, you get the original matrix back.

• For a 2x2 identity matrix, it is represented as \( I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}. \)
• Multiplying any 2x2 matrix by an identity matrix ensures that the matrix remains unchanged.

This property is useful in verifying inverse matrices, since if \( A \cdot A^{-1} = I, \)then \( A^{-1} \)is indeed the correct inverse matrix.In our solution, \[ \begin{bmatrix} 3 & 1 \ 2 & 1 \end{bmatrix}\begin{bmatrix} 1 & -1 \ -2 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}, \] confirming our calculations were accurate.

2x2 Matrices

The 2x2 matrix is one of the simplest forms of matrices, which makes it particularly easy to work with in manual calculations.
A matrix of this size has two rows and two columns, neatly organized as: \[ \begin{bmatrix} a & b \ c & d \end{bmatrix}. \]

• These matrices are used extensively in various mathematical fields, including physics and computer science.
• The simplicity of a 2x2 matrix makes it ideal for illustrating essential concepts such as determinants and inverses.

In this problem, understanding how to handle a 2x2 matrix allows us to comprehend larger and more complex matrices in advanced mathematics.
It acts as a practical introduction to matrix operations and matrix algebra.