Problem 23 - Precalculus with Limits

Question

In Exercises 13-24, find the inverse of the matrix (if it exists). \(\left[ \begin{array}{r} -8 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 4 & 0 \\\ 0 & 0 & 0 & -5 \end{array} \right]\)

Step-by-Step Solution

Answer

The inverse of the matrix is \[ \left[ \begin{array}{r} -1/8 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1/4 & 0 \ 0 & 0 & 0 & -1/5 \end{array} \right] \]

Step 1: Recognize Diagonal Matrix

This is a 4x4 diagonal matrix, meaning all its non-diagonal elements are zero. This simplifies the inversion process significantly.

Step 2: Invert Diagonal Elements

The inverse of a diagonal matrix is a diagonal matrix where each diagonal element is the reciprocal of the corresponding diagonal element of the original matrix. Therefore, we simply take the reciprocal of each diagonal element. Check that none of the diagonal elements are zero before doing so because we cannot take the reciprocal of zero.

Step 3: Formulate the Inverse Matrix

The result of taking the reciprocal of each diagonal element, \( -1/-8, 1/1, 1/4, -1/-5 \), gives us the diagonal elements of the inverse matrix. Putting them into the positions of a 4x4 matrix, we have the inverse of the original matrix.

Key Concepts

Diagonal Matrix Explained

A diagonal matrix is a matrix in which the entries outside the main diagonal are zero. Imagine a square matrix where values are positioned diagonally from the first to the last row. This diagonal is what gives such matrices their name.

• Example: Consider the diagonal matrix given: \[\begin{bmatrix} -8 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 4 & 0 \0 & 0 & 0 & -5 \end{bmatrix}\]
• Simplicity: Because all non-diagonal elements are zero, diagonal matrices are easier to work with, especially when performing operations such as finding inverses or multiplying.

The main advantage is the simplicity they bring to calculations. All focus is on the diagonal elements, significantly speeding up processes like matrix inversion.

Understanding Matrix Elements

Matrix elements are the individual numbers or variables located in a matrix, organized in rows and columns. Each element can be identified using two indices: one for the row and another for the column.

• Position Identification: In our example matrix, the element \( -8 \) is located at the first row and the first column, denoted as \( a_{11} \).
• Zero Elements: For diagonal matrices, elements outside the diagonal (like in positions \( a_{12}, a_{13} \) etc.) are zero. This feature distinguishes diagonal matrices from other types.

Focusing on elements helps us understand the matrix's structure and subsequently perform operations efficiently. For inverses, only the diagonal elements need attention, simplifying calculations further.

The Matrix Inversion Process

The matrix inversion process for diagonal matrices is straightforward due to their unique structure. Typically, inverting a matrix involves complex operations, but diagonal matrices simplify these steps.

• Inversion Steps:
  • Identify the matrix as diagonal.
  • Compute the reciprocal of each diagonal element.
  • Ensure no diagonal element is zero, as inverting would be impossible otherwise.
• Example Application: For the given matrix, the inverse involves calculating: \[\begin{align*} \frac{-1}{-8} &= \frac{1}{8}, \ \frac{1}{1} &= 1, \ \frac{1}{4} &= \frac{1}{4}, \ \frac{-1}{-5} &= \frac{1}{5} \end{align*}\]

These reciprocal values are placed back into their respective diagonal positions, forming the inverse matrix. This process highlights the efficient approach possible for diagonal matrices, which can significantly reduce computation time.