Problem 29 - A Graphical Approach to Precalculus with Limits

Question

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{lll} 2 & 1 & 2 \\ 5 & 10 & 5 \\ 3 & 6 & 3 \end{array}\right]$$

Step-by-Step Solution

Answer

Matrix \( A \) is not invertible since its determinant is 0.

Step 1: Identify the Matrix

We are given a matrix \( A \) of size \( 3 \times 3 \): \[A = \begin{bmatrix} 2 & 1 & 2 \ 5 & 10 & 5 \ 3 & 6 & 3 \end{bmatrix}\]Our task is to find the inverse of this matrix, \( A^{-1} \), if it exists.

Step 2: Check Determinant for Invertibility

To determine if \( A^{-1} \) exists, we need to check if the determinant of \( A \) is non-zero. The determinant of a \( 3 \times 3 \) matrix is calculated as:\[\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]Where the elements of the matrix correspond to:\[A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\]Substituting the values, for matrix \( A \):\[\text{det}(A) = 2(10 \times 3 - 5 \times 6) - 1(5 \times 3 - 5 \times 3) + 2(5 \times 6 - 10 \times 3)\]\[= 2(30 - 30) - 1(15 - 15) + 2(30 - 30)\]\[= 2(0) - 1(0) + 2(0)\]\[= 0\]

Step 3: Conclusion on Invertibility

Since the determinant of matrix \( A \) is 0, the matrix \( A \) does not have an inverse. A matrix is invertible only if its determinant is non-zero.

Key Concepts

Determinant Calculation

To determine whether a matrix is invertible, we must calculate its determinant. This is a special number associated with a square matrix. In the context of a 3x3 matrix, like our given matrix \( A \), the formula for the determinant is a bit complex, but it follows a consistent pattern:

• Identify and label the elements of the matrix.
• Apply the determinant formula \( \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \).
• Calculate the terms separately to avoid mistakes.

This calculation helps us understand much about the matrix, notably whether it has an inverse. When computing the determinant of our particular matrix \( A \):

• Notice that terms such as \( ei - fh \), \( di - fg \), and \( dh - eg \) are determinants of smaller 2x2 matrices.
• Given the determinant formula, substitute each matrix element into the formula meticulously. This substitution confirms the structural integrity of the matrix calculations.

In our exercise, each of these determinant expressions results in zero, leading to the conclusion that the overall determinant of matrix \( A \) is zero. This is a critical outcome in determining if the matrix can be inverted.

3x3 Matrix

A 3x3 matrix is a foundational concept in linear algebra, comprising three rows and three columns.

• The dimension, "3x3," refers to the number of rows and columns, indicating its square nature.
• This configuration is common in various applications, such as transformations and rotations in 3D space.

Matrices, especially 3x3, play crucial roles in calculations and problems. Due to their size, they allow for more complex calculations, such as finding determinants, inverses, and eigenvalues. Each element in a 3x3 matrix must be considered independently during computations, a factor making them both versatile and slightly challenging.For our specific matrix \( A \):

• The matrix is symmetrically organized, which can often hint at specific properties such as singularity or internal patterns.
• Identifying the numbers in the respective slots during calculations helps maintain accurate and precise results.

Learning to handle 3x3 matrices prepares one for intricate mathematical problems, emphasizing the significance of each step and calculation.

Matrix Invertibility

Matrix invertibility is a fundamental concept in linear algebra, mainly governed by the determinant. If the determinant of a matrix is non-zero, the matrix is invertible, meaning it has an inverse.

• An inverse matrix \( A^{-1} \) is a matrix that, when multiplied with the original matrix \( A \), produces the identity matrix \( I \).
• Only square matrices, those with equal rows and columns, like our 3x3 matrix \( A \), can potentially have inverses.

In the context of our problem, we calculated the determinant of matrix \( A \) and found it to be zero. Therefore, matrix \( A \) is not invertible. This means \( A^{-1} \) does not exist because

• A zero determinant indicates that the matrix's rows or columns are linearly dependent.
• This dependency prevents the formation of the identity matrix through multiplication.

Understanding matrix invertibility unveils why matrices can fail to have inverses and aids significantly in solving various mathematical equations and real-life modeling scenarios. It showcases the relationship between determinant calculations and the broader matrix theory.