Let's consider the $m \times n$ matrix $A$, where each column is either a zero column (all elements in the column are zero) or a column that contains all zero elements except for a single element which is equal to 1 (called a "standard basis vector"). We want to show that every minor of $A$ has a value of either 0 or $\pm 1$.

Minor of a matrix:

A minor is the determinant of a square submatrix formed by selecting certain rows and columns from the matrix. To show that every minor of $A$ has the value 0 or $\pm 1$, we will analyze how the structure of $A$ affects the determinants of these square submatrices.

Step 1: Structure of matrix $A$

Each column of $A$ falls into one of two categories:

• Zero column: A column consisting entirely of zeroes.
• Standard basis column: A column that has exactly one element equal to 1, with all other elements being zero.

Therefore, matrix $A$ is highly structured: it is composed of some zero columns and some columns that are part of the standard basis for $\mathbb{R}^m$.

Step 2: Forming a minor

When we calculate a minor, we extract a square submatrix by selecting certain rows and columns from $A$. The value of this minor depends on which columns we select:

• If one of the selected columns is a zero column, the determinant of the submatrix will be zero because that column contributes no non-zero entries.
• If none of the selected columns are zero columns, each column will be a standard basis vector.

Step 3: Determinant of a submatrix of standard basis vectors

If the selected columns are standard basis vectors, the submatrix will have a particularly simple structure:

• Each column has exactly one element equal to 1, with all other elements equal to 0.
• Each row and column of the submatrix will have exactly one entry equal to 1, and all other entries will be 0, which corresponds to a permutation matrix.

The determinant of a permutation matrix is either $+1$ or $-1$, depending on whether the corresponding permutation is even or odd. Therefore, if the submatrix consists entirely of standard basis columns, the determinant will be either $+1$ or $-1$.

Step 4: Conclusion

• If any zero column is included in the minor, the determinant of the submatrix will be 0.
• If all columns are standard basis columns, the submatrix will be a permutation matrix, and its determinant will be either $+1$ or $-1$.

Thus, every minor of $A$ has a determinant that is either $0$, $+1$, or $-1$.

Do you want more details or have any questions?

Here are 5 questions related to this topic:

1. How does the structure of a matrix influence the value of its minors?
2. What is a standard basis vector, and how does it affect matrix minors?
3. Can you find examples of matrices where all minors are zero?
4. How does the determinant of a permutation matrix relate to the sign of a permutation?
5. What other matrix types have minors restricted to specific values?

Tip: Understanding the properties of permutation matrices is essential in grasping how determinants of structured matrices behave.