Consider an m×n matrix A such that every column of A contains either
only zero elements or zero elements throughout except one which has
the value unity. Show that every minor of A has either has the value 0
or ±1.

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.

Consider an m×n matrix A such that every column of A contains either only zero elements or zero elements throughout except one which has the value unity. Show that every minor of A has either the value 0 or ±1.

Discover how to prove that every minor of an m×n matrix, where each column is either a zero column or a standard basis vector, has a determinant of 0 or ±1. This solution covers matrix minors, determinants, and permutation matrices, suitable for undergraduate-level students.

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