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GO Classes CS/DA 2025 | Weekly Quiz 5 | Linear Algebra | Question: 1

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Let $A$ be an $n \times n$ matrix of real or complex numbers. Which of the following statements are equivalent to: “the matrix $A$ is invertible”?

  1. The columns of $A$ are linearly independent.
  2. The rows of $A$ are linearly independent.
  3. The only solution of the homogeneous equations $Ax = 0$ is $x = 0$.            
  4. The rank of $A$ is $n$.
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2 Answers

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If matrix is invertible then determinant of matrix is always non zero, i.e. rows and column of the matrix are linearly independent.
 
If $Ax=0$ has only a unique solution then $A$ does not have any free variable. So the determinant of $A$ can’t be zero hence $A$ is invertible.

All columns are linearly dependent thus rank is $n$.
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->As the information given to us :

"the matrix A is invertible"

-> we know that matrix x will be invertible only if the matrix itself is linearly independent, so in this case Matrix A is linear dependent,

-> which means its rows, columns are linearly dependent, (A,B).

-> If matrix is L.I. than in the equation Ax=0, matrix A can never be zero hence x =0 is also true(C)

-> as we have Matrix A is L.I. hence rank of A is number of its columns or no. rows in this case it is n(D)
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