1. Prove: If B2×2 is a scalar multiple of I2, then AB = BA ∀A2×22. Let An×n be upper triangular. Prove by induction that |A|= ∏ni=1 aii3. Consider the matrix A = I −X(X′X)−1X′. Prove that A ·A = A. What restrictions need to be imposed on X and (X′X) for this to hold? 1.Prove: If B₂x2 is a scalar multiple of I2, then AB = BA VA₂×22.Let Anxn be upper triangular. Prove by induction that |A| = [[?_1 Aiii=13.-Consider the matrix A = I – X(X'X)-¹X'. Prove that A . A = A. What restrictions need to be imposedon X and (X'X) for this to hold?

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Math

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If B2×2 is a scalar multiple of I2, then AB = BA ∀A2×2

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

1. Prove: If B2×2 is a scalar multiple of I2, then AB = BA ∀A2×2

2. Let An×n be upper triangular. Prove by induction that |A|= ∏ni=1 aii

3. Consider the matrix A = I −X(X′X)−1X′. Prove that A ·A = A. What restrictions need to be imposed on X and (X′X) for this to hold?

1.
Prove: If B₂x2 is a scalar multiple of I2, then AB = BA VA₂×2
2.
Let Anxn be upper triangular. Prove by induction that |A| = [[?_1 Aii
i=1
3.
-
Consider the matrix A = I – X(X'X)-¹X'. Prove that A . A = A. What restrictions need to be imposed
on X and (X'X) for this to hold?

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