Problem 50 - College Algebra

Question

For the following exercises, fi d the inverse of the given matrix. $$\left[\begin{array}{lllll}1 & 2 & 0 & 2 & 3 \\ 0 & 2 & 1 & 0 & 0 \\ 0 & 0 & 3 & 0 & 1 \\ 0 & 2 & 0 & 0 & 1 \\ 0 & 0 & 1 & 2 & 0\end{array}\right]$$

Step-by-Step Solution

Answer

Check if the matrix determinant is non-zero, then use Gaussian elimination or adjugate method to find the inverse.

Step 1: Verify Matrix Dimensions

First, confirm the matrix is square by checking its dimensions. The given matrix has dimensions 5x5, making it a square matrix, which is a requirement for finding the inverse.

Step 2: Check Matrix Non-Singularity

To find the inverse, the matrix must be non-singular, meaning its determinant should not be zero. Calculate the determinant of the given matrix to verify this condition.

Step 3: Calculate the Determinant

Use cofactor expansion along an appropriate row or column due to the zeros in the matrix to simplify calculations. Calculate the determinant and ensure it is non-zero.

Step 4: Apply Row Reduction or Adjoint Method

With the determinant confirmed non-zero, use the row reduction (Gaussian elimination) or the adjugate and determinant method to find the inverse. This involves transforming the matrix into the identity matrix while applying the same operations to an identity matrix of the same size.

Step 5: Validate the Inverse

Multiply the original matrix by its computed inverse to verify that the result is an identity matrix. This confirms the correctness of the inverse matrix found.

Key Concepts

Square Matrix

A square matrix is a type of matrix that has the same number of rows and columns. It can be represented as an \( n \times n \) matrix, where n is a positive integer.

This characteristic is essential when finding the inverse of a matrix because only square matrices can potentially have inverses. The matrix given in the exercise is a 5x5 square matrix.

This feature of having equal rows and columns plays a critical role in operations like determinant calculation and Gaussian elimination. Both of these methods rely on manipulating square matrices to perform calculations like finding inverses.

Determinant Calculation

Before finding the inverse of a matrix, it's crucial to compute its determinant. The determinant is a special number that can be calculated from a square matrix. It provides critical information about the matrix, including whether the matrix is invertible.

If the determinant of a square matrix is zero, the matrix is considered singular, meaning it does not have an inverse. Conversely, a non-zero determinant indicates that the matrix is non-singular and an inverse can be calculated.

• To calculate the determinant of a 5x5 matrix like the one in our exercise, you can use cofactor expansion. Choosing a row or column with the most zeros can simplify the process.
• Remember, calculating the determinant involves multiplying across the diagonal and subtracting cross-products, which become complex in larger matrices.

Finding the determinant is a necessary step to determine if proceeding with inverse calculations is possible.

Gaussian Elimination

Gaussian elimination is a method used to simplify matrices, often used to find their inverse. This method is a systematic technique for performing row operations to transform a matrix into its reduced row-echelon form.

In the process, the goal is to use row operations to convert the original matrix into an identity matrix. Meanwhile, identical operations are applied to a separate identity matrix, which eventually transforms into the inverse if the matrix is non-singular.

• The steps include swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting multiples of one row from another row.
• When the original matrix reaches the identity form, and the separate identity matrix has transformed alongside these changes, the latter becomes the matrix's inverse.

This method is comprehensive and works well for finding inverses of larger square matrices like a 5x5.

Adjugate Matrix

Another method for finding the inverse of a matrix involves using its adjugate. The adjugate matrix is derived from the original matrix and is useful in the context of matrix inversion.

Specifically, if you have a square matrix A, its inverse A\(^{-1}\) can be found using the formula: \ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \, where \( \text{adj}(A) \) is the adjugate of A and \( \text{det}(A) \) is the determinant of A.

• The adjugate of a matrix is obtained by taking the transpose of its cofactor matrix. Each element of the cofactor matrix is the determinant of a smaller matrix, or minor, after removing the row and column of the element being replaced.
• This method is particularly useful if the determinant is already known and non-zero, as it then simplifies the calculation of the inverse.

Understanding how to form and use the adjugate matrix can offer a more straightforward path to inverting matrices, especially useful with smaller-sized matrices.