Problem 28 - A Graphical Approach to Precalculus with Limits

Question

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{rrr} 0.8 & 0.2 & 0.1 \\ -0.2 & 0 & 0.3 \\ 0 & 0 & 0.5 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix does not have an inverse because its determinant is 0.

Step 1: Verify if the Matrix is Square

Check that the given matrix \( A \) is a square matrix. Since the matrix is a \( 3 \times 3 \) matrix (3 rows and 3 columns), it is indeed a square matrix.

Step 2: Check the Determinant of the Matrix

Calculate the determinant of the matrix \( A \) using the formula for a \( 3 \times 3 \) matrix:\[\text{det}(A) = 0.8(0 \cdot 0.5 - 0.3 \cdot 0) - 0.2(-0.2 \cdot 0.5 - 0.3 \cdot 0) + 0.1(-0.2 \cdot 0 - 0 \cdot 0) = 0\]The determinant of the matrix \( A \) is 0.

Step 3: Analyze the Matrix Invertibility

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

Key Concepts

Square Matrix

In mathematics, a square matrix is a matrix with the same number of rows and columns. This symmetry makes it essential in many linear algebra operations such as calculating the determinant and finding an inverse if possible. For example, a matrix with three rows and three columns is referred to as a 3x3 matrix.
Square matrices can be identified easily by looking at their dimensions. If you are told a matrix is 2x2, 3x3, or even 10x10, you know it is square. Their structure is fundamental in many applications, including systems of equations and transformations.
One key property of square matrices is that they can potentially have an inverse, allowing for the solving of linear equations. However, not all square matrices are invertible, and checking the determinant often provides this necessary information.

Determinant

The determinant is a special number calculated from a square matrix. It provides vital information about the matrix, such as whether the matrix is invertible or not. For a 3x3 matrix, the determinant involves a specific formula involving multiplication and addition of its elements.
For a matrix \[A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\]The determinant is computed as:\[det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\]
Calculating this value helps in understanding properties like eigenvalues and the matrix's invertibility. If the determinant is zero, it indicates a singular matrix, meaning the matrix does not have an inverse. Understanding the determinant and how to calculate it is crucial for students diving into linear algebra.

Invertibility

Invertibility is a property of square matrices where the matrix has an inverse. The inverse of a matrix, denoted as \(A^{-1}\), is a matrix that, when multiplied by the original matrix \(A\), results in the identity matrix.
For a matrix to be invertible, its determinant must be non-zero. In mathematical terms, if \[det(A) eq 0\]then \(A\) is invertible. This concept helps in solving systems of linear equations, finding power series solutions, and performing matrix transformations.
If you're tackling a matrix and hoping to find its inverse, always begin by calculating its determinant. If the determinant is zero, as shown in the example, then the matrix is singular, which implies it's not invertible. Understanding these conditions is a crucial step in mastering matrix operations.

3x3 Matrix

A 3x3 matrix is a matrix with three rows and three columns. This size of matrix is often explored in linear algebra due to its balance between simplicity and complexity, allowing students to understand basic and more advanced concepts.
An example of a 3x3 matrix is:\[\begin{bmatrix} 0.8 & 0.2 & 0.1 \ -0.2 & 0 & 0.3 \ 0 & 0 & 0.5 \end{bmatrix}\]
Students frequently use such matrices to grasp core concepts like calculating determinants, understanding invertibility, and performing operations such as matrix multiplication and finding eigenvalues. Each entry of the matrix is vital for operations, making them a cornerstone in matrices studies across many different academic disciplines.
When analyzing a 3x3 matrix, understanding its internal structure and how each element interacts with others in equations, especially when calculating something like a determinant, will streamline many mathematics problems and tasks.