Problem 11 - College Algebra and Calculus: An Applied Approach

Question

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{ll} 1 & 2 \\ 3 & 7 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of the given matrix is \(A^{-1} = \begin{bmatrix} 7 & -2 \\ -3 & 1 \end{bmatrix}\).

Step 1: Compute the determinant

The determinant of the matrix \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) is computed by the formula \(ad - bc\). In this case it's \(1*7 - 2*3 = 7 - 6 = 1\). As the determinant isn't zero, it confirms that the inverse of the matrix exists.

Step 2: Apply the inverse formula

The inverse of a 2x2 matrix \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) can be found using the formula \(A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\). Apply this formula to get the inverse as \(A^{-1} = \begin{bmatrix} 7 & -2 \\ -3 & 1 \end{bmatrix}\)

Step 3: Confirm the result

The inverse matrix should satisfy the condition that when it's multiplied by the original matrix, the result should be an identity matrix. Let's verify this: \(A * A^{-1} = \begin{bmatrix} 1 & 2 \\ 3 & 7 \end{bmatrix} * \begin{bmatrix} 7 & -2 \\ -3 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) which verifies our calculation.

Key Concepts

Determinant

The concept of a determinant is crucial for understanding if a matrix inversion is possible. In simple terms, the determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix, which is a matrix with 2 rows and 2 columns, the determinant helps us decide if the matrix can be inverted. To find the determinant of a matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\),you use the formula:\[ \text{det}(A) = ad - bc\]This calculation involves multiplying the elements on the diagonal from top left to bottom right (\(a\) and \(d\)) and subtracting the product of the elements on the other diagonal (\(b\) and \(c\)). A key takeaway is:

• If the determinant is not 0, the matrix can be inverted.
• If the determinant is 0, then the matrix does not have an inverse.

In our example, the determinant of the matrix \(\begin{bmatrix} 1 & 2 \ 3 & 7 \end{bmatrix}\) is 1, confirming the inverse exists.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of matrices that still allows us to perform interesting operations like inversion. Its structure is perfect for beginners to understand basic operations applicable to matrices. The 2x2 matrix is written as:\[ \begin{bmatrix} a & b \ c & d \end{bmatrix}\]where:

• \(a\),\(b\),\(c\), and \(d\) are elements of the matrix.
• Each of these elements can be a number that comes from solving equations.

When you want to find the inverse of a 2x2 matrix, you rely on both its elements and the determinant. The formula used for inverting is:\[A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]Here:

• The original elements \(a\), \(b\), \(c\), and \(d\) swap their positions and signs.
• Importantly, you must divide the entire matrix by the determinant.

This easy manipulation of a 2x2 matrix helps to grasp how matrix inversions work fundamentally.

Matrix Multiplication

Matrix multiplication is a fundamental operation used to verify the correctness of a matrix inversion. When you multiply a matrix by its inverse, you should get the identity matrix as a result.For a 2x2 matrix, let's express the operation with two matrices \(A = \begin{bmatrix} 1 & 2 \ 3 & 7 \end{bmatrix}\) and \(A^{-1} = \begin{bmatrix} 7 & -2 \ -3 & 1 \end{bmatrix}\).The multiplication to check for correctness follows this way:1. Multiply the elements row-wise and column-wise: \[\begin{bmatrix} 1\cdot7 + 2\cdot(-3) & 1\cdot(-2) + 2\cdot1 \ 3\cdot7 + 7\cdot(-3) & 3\cdot(-2) + 7\cdot1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]2. The result is the identity matrix.

• The identity matrix has ones on its diagonal and zeros elsewhere.
• It effectively verifies that the inverse matrix calculation was done correctly.

Understanding this process cements your knowledge of how matrices and their inverses interact through multiplication.