Problem 51 - College Algebra with Corequisite Support

Question

For the following exercises, find the inverse of the given matrix. $$ \left[\begin{array}{llllll} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 \end{array}\right] $$

Step-by-Step Solution

Answer

The matrix is not invertible due to linearly dependent rows.

Step 1: Understand the Matrix

The given matrix is a 6x6 matrix. It consists mostly of an identity matrix with all 1s in the final row: \[\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \0 & 1 & 0 & 0 & 0 & 0 \0 & 0 & 1 & 0 & 0 & 0 \0 & 0 & 0 & 1 & 0 & 0 \0 & 0 & 0 & 0 & 1 & 0 \1 & 1 & 1 & 1 & 1 & 1 \\end{bmatrix}\]

Step 2: Establish the Criteria for Invertibility

Only square matrices with linearly independent rows (or columns) can have inverses. The row of all 1s indicates linear dependence with the other rows, violating the criteria for invertibility.

Step 3: Conclude Non-Invertibility

Since the matrix has a row of all 1s, which is a linear combination of other unit vectors, the matrix is not invertible. An invertible matrix cannot have linearly dependent rows.

Key Concepts

Invertible Matrix

An invertible matrix, sometimes referred to as a non-singular matrix, is a matrix that possesses what is called an inverse. To better grasp this, consider an inverse as a sort of 'reverse' operation for matrices. When multiplied by its inverse, the result is the identity matrix, which we'll talk about later. An essential quality of an invertible matrix is being square, meaning it has the same number of rows and columns. What makes a matrix invertible?

• The matrix must be square.
• Its rows (or columns) should be linearly independent.
• The determinant of the matrix should not be zero.

If a matrix fails to meet these criteria, such as when rows are linearly dependent, it does not have an inverse, making it non-invertible.

Linear Dependence

Linear dependence is a concept from linear algebra that determines how vectors relate to each other. If you can express one vector as a combination of others, they are linearly dependent. In the context of a matrix, if a row can be formed by adding, subtracting, or multiplying other rows, then the rows are linearly dependent. Why does this matter for invertibility?

• Linearly dependent rows indicate a lack of unique solutions.
• Such rows lead to the determinant being zero, which is a key factor for invertibility.
• If any row of a matrix is a linear combination of other rows, the matrix fails the invertibility test.

In our given 6x6 matrix example, the last row is made up of all 1s, which is indeed a linear combination of the previous rows. Therefore, the matrix is linearly dependent and not invertible.

Identity Matrix

The identity matrix is a special type of square matrix. It behaves similarly to the number 1 in multiplication; that is, any number multiplied by 1 remains unchanged. When a matrix is multiplied by the identity matrix, the original matrix is what you get back.Key features of the identity matrix include:

• Being square: it has equal numbers of rows and columns.
• Diagonal elements are 1s, and all other elements are 0s.
• For a matrix with dimension \( n \times n \), the identity matrix is denoted as \( I_n \).

In the provided exercise, the top 5 rows of the matrix resemble an identity matrix, as they have 1s down the diagonal. However, the last row deviates from this structure, making it not a full identity matrix. Despite appearing close, these changes prevent it from maintaining the properties necessary for the overall matrix to be invertible.