Problem 26 - College Algebra with Corequisite Support

Question

For the following exercises, find the multiplicative inverse of each matrix, if it exists. $$ \left[\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right] $$

Step-by-Step Solution

Answer

The matrix does not have a multiplicative inverse because its determinant is zero.

Step 1: Check if the matrix is square

A matrix must be square (same number of rows and columns) to potentially have an inverse. The given matrix is 3x3, which is square, so it may have an inverse.

Step 2: Calculate the determinant

To find the inverse of a matrix, its determinant must be non-zero. The determinant of a 3x3 matrix \(A\) with elements \(a_{ij}\) is given by:\[\text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\]For the matrix:\[A = \begin{pmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{pmatrix}\]Plug the values into the determinant formula:\[\text{det}(A) = 1(5 \times 9 - 6 \times 8) - 2(4 \times 9 - 6 \times 7) + 3(4 \times 8 - 5 \times 7)\]Calculating gives:\[\text{det}(A) = 1(45 - 48) - 2(36 - 42) + 3(32 - 35)\]\[= 1(-3) - 2(-6) + 3(-3)\]\[= -3 + 12 - 9\]\[= 0\]

Step 3: Conclusion based on the determinant

Since the determinant of the matrix is zero, the matrix does not have a multiplicative inverse. A matrix with a determinant of zero is called singular and cannot be inverted.

Key Concepts

Determinant

The determinant is a value that can be calculated from a square matrix. It is a crucial part of matrix algebra and plays a significant role in determining the invertibility of a matrix. For a 3x3 matrix, the determinant can express the volume scaling factor of a transformation described by the matrix.
The formula for the determinant of a 3x3 matrix is a bit more complex than smaller matrices. It is calculated as follows:

• Start with the first row, and take the element in consideration, say, element 1.
• Multiply this by the determinant of the 2x2 matrix left after removing the row and column containing this element.
• Repeat this process for the remaining elements of the first row, ensuring that alternating signs are used.

In this exercise, the determinant was calculated as:\[\text{det}(A) = 1\times(5\times9 - 6\times8) - 2\times(4\times9 - 6\times7) + 3\times(4\times8 - 5\times7) = 0\]Since the determinant is zero, the matrix is singular, which we'll explain more about in the below section.

Matrix Algebra

Matrix algebra is a field of mathematics that involves the study of matrices and their operations. It offers various techniques for manipulating matrices, such as addition, multiplication, and finding inverses.
One of the significant operations in matrix algebra is finding the inverse of a matrix. The inverse of a matrix \(A\) exists if and only if certain conditions are met:

• The matrix must be square (same number of rows and columns).
• The determinant of the matrix must be non-zero.

When a matrix has an inverse, the product of the matrix and its inverse results in the identity matrix, which acts like 1 in regular algebra. For example, \(A \times A^{-1} = I\), where \(I\) is the identity matrix.
In the exercise, we established that the matrix is square since it has dimensions of 3x3. However, due to its zero determinant, it does not have an inverse. Understanding these properties is crucial for dealing with matrix-based problems.

Singular Matrix

A singular matrix is a square matrix that does not have an inverse. The primary characteristic of a singular matrix is that its determinant is zero. This property disallows the matrix from having any unique solutions or being inverted. In practical terms, singular matrices can indicate that the system of equations represented by the matrix cannot be solved uniquely.
In the given problem, we found that the determinant of the matrix is zero, indicating that it is a singular matrix:\[\text{det}(A) = 0\]This matches the definition of a singular matrix and tells us that there is no possible inverse to be computed. Recognizing whether a matrix is singular is vital, especially in solving systems of equations, as it tells whether a unique solution exists or not.