Math Problem Statement

Let š“ be an invertible square matrix. It is possible for š“ to have a column of zeros.

Question 1 Select one: True False

Solution

The correct answer is False.

If AA is an invertible square matrix, it cannot have a column (or row) of zeros. This is because if any column (or row) is a zero vector, the determinant of the matrix would be zero, and a matrix with a zero determinant is not invertible.

Therefore, an invertible matrix must have full rank, meaning no columns or rows can be linearly dependent (including having a column of zeros).

Would you like further explanation or have more questions on this topic?

Here are 5 related questions for you:

  1. What is the condition for a matrix to be invertible?
  2. How do you calculate the determinant of a matrix?
  3. What does it mean for a matrix to be of full rank?
  4. How does a column of zeros affect the rank of a matrix?
  5. Can a matrix with a row of zeros be invertible?

Tip: To check if a matrix is invertible, always calculate its determinant—if it's non-zero, the matrix is invertible!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Inversion
Determinants

Formulas

Determinant of a matrix
Condition for matrix invertibility (det(A) ≠ 0)

Theorems

Invertible matrix theorem
Full rank condition for invertibility

Suitable Grade Level

Grades 11-12