Problem 13 - College Algebra

Question

For the following exercises, find the multiplicative inverse of each matrix, if it exists. \(\left[\begin{array}{rr}3 & -2 \\ 1 & 9\end{array}\right]\)

Step-by-Step Solution

Answer

The inverse is \( \begin{bmatrix} \frac{9}{29} & \frac{2}{29} \\ -\frac{1}{29} & \frac{3}{29} \end{bmatrix} \).

Step 1: Determine if the Matrix is Invertible

To check if the matrix is invertible, we must first find its determinant. The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated as \( ad - bc \). For the given matrix \([\begin{array}{rr}3 & -2 \ 1 & 9\end{array}]\), the determinant is \(3 \cdot 9 - (-2) \cdot 1 = 27 + 2 = 29\). Since the determinant is not zero, the matrix is invertible.

Step 2: Find the formula for the Inverse Matrix

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the inverse is given by \( \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). Substitute \(a = 3\), \(b = -2\), \(c = 1\), and \(d = 9\) into this formula.

Step 3: Calculate the Inverse Matrix

Using the values from Step 2 and the determinant from Step 1, the inverse matrix is calculated as: \( \frac{1}{29} \begin{bmatrix} 9 & 2 \ -1 & 3 \end{bmatrix} \). Distribute the \(\frac{1}{29}\) to each element within the matrix: \[\begin{bmatrix} \frac{9}{29} & \frac{2}{29} \ -\frac{1}{29} & \frac{3}{29} \end{bmatrix}\].

Step 4: Verify the Inverse (Optional, for Understanding)

To confirm the solution, multiply the original matrix by its inverse. If the result is the identity matrix \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \), the inverse is correct. For our matrix, multiply: \( \begin{bmatrix} 3 & -2 \ 1 & 9 \end{bmatrix} \begin{bmatrix} \frac{9}{29} & \frac{2}{29} \ -\frac{1}{29} & \frac{3}{29} \end{bmatrix} \), verifying it results in the identity matrix.

Key Concepts

Determinants

The determinant is a special number that can be calculated from a square matrix, and it plays a crucial role in determining whether a matrix is invertible. For a 2x2 matrix, it's quite simple to calculate: given the matrix structure \[\begin{bmatrix}a & b \c & d\end{bmatrix}\]the determinant \(det\) can be found using the formula \(ad - bc\).

• The determinant provides insight into the matrix's properties. A non-zero determinant indicates the matrix has an inverse, making it invertible.
• If the determinant is zero, the matrix is not invertible, also known as singular.

For our matrix \[\begin{bmatrix}3 & -2 \1 & 9\end{bmatrix}\]calculating the determinant involves substituting the respective numbers into the formula: \(3 \cdot 9 - (-2) \cdot 1 = 27 + 2 = 29\). Since this result is not zero, we conclude the matrix is invertible, allowing us to proceed to find its inverse.

Invertible Matrices

An invertible matrix, also known as a non-singular matrix, is a square matrix that has an inverse. The concept of matrix inverses is pivotal in linear algebra as it facilitates solving linear equations, among other applications. A matrix \(A\) is invertible if there exists another matrix \(A^{-1}\) such that when \(A\) is multiplied by \(A^{-1}\), the product is the identity matrix.In simpler terms:

• For a matrix to be invertible, its determinant must be non-zero.
• The inverse of a 2x2 matrix \[\begin{bmatrix}a & b \c & d\end{bmatrix}\]is calculated using the formula \[\frac{1}{ad - bc} \begin{bmatrix}d & -b \-c & a\end{bmatrix}.\]

The existence of the inverse allows us to solve matrix equations of the form \(AX = B\) by multiplying both sides by the inverse matrix \(A^{-1}\), thus \(X = A^{-1}B\). It's important to note that not every matrix is invertible, emphasizing the determinant's indispensable role in this determination.

2x2 Matrices

2x2 matrices are perhaps the simplest form of square matrices. They consist of two rows and two columns, making them manageable to work with and a great starting point for learning about matrix operations such as inverse calculation, determinant determination, and basic multiplication.Here's a quick breakdown of their properties:

• They have a straightforward structure: two rows and two columns, represented as \[\begin{bmatrix}a & b \c & d\end{bmatrix}\].
• Operations on 2x2 matrices, like finding determinants and inverses, are computationally less intensive compared to larger matrices.
• The identity matrix for a 2x2 system is always \[\begin{bmatrix}1 & 0 \0 & 1\end{bmatrix}\], which serves as a crucial element when verifying inverse calculations.

Given how fundamental 2x2 matrices are, mastering them aids significantly in grasping more complex matrix concepts and operations later on. They serve as an excellent entry point into the broader and more intricate field of linear algebra.