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 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:
- What is the condition for a matrix to be invertible?
- How do you calculate the determinant of a matrix?
- What does it mean for a matrix to be of full rank?
- How does a column of zeros affect the rank of a matrix?
- Can a matrix with a row of zeros be invertible?
Tip: To check if a matrix is invertible, always calculate its determinantāif 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) ā 0)
Theorems
Invertible matrix theorem
Full rank condition for invertibility
Suitable Grade Level
Grades 11-12
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