What is A2 when N(A) = C(A) and the columns of A are in the nullspace of A?

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SUMMARY

When a matrix A satisfies the condition N(A) = C(A), it indicates that A is a square matrix and its rank is equal to its nullity. Consequently, A² results in the zero matrix. This conclusion arises from the fact that if the columns of A are in the nullspace of A, then multiplying A by itself leads to A*A = 0. Thus, A² = 0 is established as a definitive outcome.

PREREQUISITES
  • Understanding of linear algebra concepts such as nullspace and column space.
  • Familiarity with matrix operations, specifically matrix multiplication.
  • Knowledge of the Rank-Nullity Theorem.
  • Basic comprehension of square matrices and their properties.
NEXT STEPS
  • Study the Rank-Nullity Theorem in detail.
  • Explore properties of square matrices and their implications.
  • Learn about the implications of nullspace and column space in linear transformations.
  • Investigate examples of matrices that satisfy N(A) = C(A) and their behavior under multiplication.
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as data scientists and engineers working with matrix computations.

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You have a matrix A such that N(A) = C(A), what can you say about A2? Why?
 
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if the columns of A is in the nullspace of A, what is the answer to A*A?
 

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