Problem 4 - College Algebra with Corequisite Support

Question

Can a matrix with an entire column of zeros have an inverse? Explain why or why not.

Step-by-Step Solution

Answer

A matrix with an entire column of zeros cannot have an inverse because its determinant is zero.

Step 1: Understanding the Inverse Matrix

An inverse matrix for a given square matrix \( A \) is another matrix, denoted as \( A^{-1} \), such that when they are multiplied together, they yield the identity matrix. Specifically, \[A \cdot A^{-1} = A^{-1} \cdot A = I\]where \( I \) is the identity matrix, which has 1s along the diagonal and 0s elsewhere.

Step 2: Condition for Invertibility

A square matrix is invertible (non-singular) if and only if its determinant is non-zero. If the determinant of a matrix is zero, then the matrix is singular, meaning it does not have an inverse.

Step 3: Determinant and Matrix with Zero Column

If a square matrix has an entire column of zeros, it implies that the matrix transformation compresses space into a lower dimension. In simpler terms, the matrix cannot span the entire space of its size, leading to a determinant of zero.

Step 4: Conclusion

Since a matrix with a column of zeros has a determinant of zero, it is singular and does not have an inverse. This is because it cannot satisfy the condition \( A \cdot A^{-1} = I \) as there's no way to retrieve the identity matrix from such a transform.

Key Concepts

Determinant

To understand why a matrix with an entire column of zeros cannot have an inverse, we first need to discuss the determinant. The determinant is a special number that can be calculated from a square matrix. It has some important properties:

• It helps determine if a matrix has an inverse. If the determinant is zero, the matrix does not have an inverse.
• The determinant tells us about the volume scaling factor of the linear transformation described by the matrix. Essentially, if the determinant is zero, it means the transformation squashes the space into a lower dimension.

When a matrix includes a whole column (or row) of zeros, the determinant becomes zero. This shows that the matrix cannot properly expand space, which means it can't return to the original space through an inverse transformation.

Identity Matrix

The identity matrix is a key concept when talking about inverses. It's essentially the 'do nothing' matrix in matrix operations.

• It is a square matrix, meaning it has the same number of rows and columns.
• The diagonal elements are all 1s, and all off-diagonal elements are 0s.
• When any matrix is multiplied by the identity matrix, it remains unchanged. Mathematically, for any matrix \( A \), \( A \cdot I = I \cdot A = A \).

In terms of inverses, the identity matrix is what you get when you multiply a matrix by its inverse. This relationship is crucial because only invertible (non-singular) matrices can achieve this product. If a matrix has a column of zeros, it fails to be invertible because it can't produce the identity matrix through multiplication with any other matrix.

Singular Matrix

A singular matrix is one that does not have an inverse.

• This happens when the determinant of a matrix is zero.
• Singular matrices often arise when there is some degeneration, like when a column is entirely zeros, indicating a loss of dimensionality.
• Without an inverse, solving matrix equations becomes impossible since you can't 'unwrap' a transformation to find original inputs.

Understanding that a singular matrix, like one with a column of zeros, signifies a collapse of the transformation matrix helps to visualize why inverses cannot exist for such matrices. They simply lack the full dimensional breadth to return back to the original matrix form needed to attain the identity matrix.