Taking L.H.S A^(k+1) = A^k . A = ... thus P(k + 1) is true. By the principle of mathematical induction, P(n) is true for n ∈ N

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Understand the Problem

The question is discussing the use of mathematical induction to establish the validity of a matrix equation. It shows the steps taken to prove that if a statement P(k) is true, then P(k+1) is also true, leading to the conclusion that P(n) is valid for all natural numbers n.

Answer

The statement \( P(n) \) is true for all natural numbers \( n \).
Answer for screen readers

The equation is proven true for all natural numbers ( n ) using mathematical induction.

Steps to Solve

  1. Setting up the Left-Hand Side (LHS) We start with the expression for ( A^{k+1} ): $$ A^{k+1} = A^k \cdot A = \begin{bmatrix} 1 + 2k & -4k \ 1 - 2k & 1 \end{bmatrix} \cdot \begin{bmatrix} 3 & -4 \ 1 & -1 \end{bmatrix} $$

  2. Matrix Multiplication Multiply the two matrices: $$ = \begin{bmatrix} (1 + 2k)3 + (-4k)(1) & (1 + 2k)(-4) + (-4k)(-1) \ (1 - 2k)3 + 1(-4) & (1 - 2k)(-4) + 1(-1) \end{bmatrix} $$

  3. Calculating Each Element Calculate each element in the resulting matrix:

    • First row, first element: $$ 3 + 6k - 4k = 3 + 2k $$
    • First row, second element: $$ -4 - 8k + 4k = -4 - 4k $$
    • Second row, first element: $$ 3 - 6k - 4 = -1 - 2k $$
    • Second row, second element: $$ -4 + 2k - 1 = -5 + 2k $$
  4. Representing the Result The resulting matrix is: $$ \begin{bmatrix} 3 + 2k & -4 - 4k \ -1 - 2k & -5 + 2k \end{bmatrix} $$

  5. Showing it Equals the Right-Hand Side (RHS) We need to show: $$ \begin{bmatrix} 1 + 2(k+1) & -4(k+1) \ 1 + k + 1 & 1 - 2(k+1) \end{bmatrix} $$ This simplifies to: $$ \begin{bmatrix} 1 + 2k + 2 & -4k - 4 \ 1 + k + 1 & 1 - 2k - 2 \end{bmatrix} = \begin{bmatrix} 3 + 2k & -4 - 4k \ 2 + k & -1 - 2k \end{bmatrix} $$

  6. Conclusion of Induction Step Since LHS equals RHS, we conclude: ( P(k+1) ) is true, thus by induction ( P(n) ) holds for all ( n \in \mathbb{N} ).

The equation is proven true for all natural numbers ( n ) using mathematical induction.

More Information

This showcases the principle of mathematical induction in proving matrix properties, essential in linear algebra.

Tips

  • Forgetting to correctly perform matrix multiplication can lead to incorrect results.
  • Miscalculating terms when simplifying algebraic expressions.

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