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GATE 2023 ST – Question 13

Matrix Theory · Vector Spaces and Linear Systems · 1 mark · Multiple choice

For real n×n matrices: I:if B is invertible,rank(BA)=rank(A). II:if A²x=b is solvable for every b∈ℝⁿ,then Ax=b is solvable for every b. Which hold?

  1. Only I
  2. Only II
  3. Both I and II
  4. Neither I nor II

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Correct answer: (C) Both I and II

Explanation

Left multiplication by an invertible matrix is an isomorphism on the column space and preserves rank, proving I. In II,A² is onto; its image lies in the image of A,so A is onto as well. Both hold. **C**.

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