Math Problem Statement

6. Find all 2 x 2 matrices which commute with the matrix [ [2, 3], [1, 4] ]. 7. If A = [ [1, 0], [-1, 1] ], show that A^2 - 2A + I_2 = O_2. Hence find A^50.

Solution

Let's solve these problems one by one.

Problem 6

Find all 2×22 \times 2 matrices that commute with the matrix [2314].\begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}.

To find all matrices B=[abcd]B = \begin{bmatrix} a & b \\ c & d \end{bmatrix} that commute with A=[2314]A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}, we need: AB=BA.AB = BA.

Expanding both ABAB and BABA and equating the entries will yield a system of equations in terms of aa, bb, cc, and dd.

  1. Compute ABAB: AB=[2314][abcd]=[2a+3c2b+3da+4cb+4d].AB = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} 2a + 3c & 2b + 3d \\ a + 4c & b + 4d \end{bmatrix}.

  2. Compute BABA: BA=[abcd][2314]=[2a+b3a+4b2c+d3c+4d].BA = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} = \begin{bmatrix} 2a + b & 3a + 4b \\ 2c + d & 3c + 4d \end{bmatrix}.

Equating ABAB and BABA:

2a + 3c = 2a + b, \\ 2b + 3d = 3a + 4b, \\ a + 4c = 2c + d, \\ b + 4d = 3c + 4d. \end{cases}$$ Solving this system will give the set of matrices that commute with $$A$$. --- ### Problem 7 **If $$A = \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix}$$, show that $$A^2 - 2A + I_2 = O_2$$. Hence, find $$A^{50}$$.** 1. **Calculate $$A^2$$:** $$A^2 = \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix} = A.$$ 2. **Substitute into $$A^2 - 2A + I_2$$:** $$A^2 - 2A + I_2 = A - 2A + I_2 = -A + I_2 = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = O_2.$$ This shows that $$A$$ satisfies the equation $$A^2 - 2A + I_2 = O_2$$, meaning $$A$$ is idempotent. 3. **Find $$A^{50}$$:** Since $$A = A^2$$, $$A^n = A$$ for any $$n \geq 1$$. Therefore, $$A^{50} = A.$$ --- Would you like further explanations or details on any part of this solution? --- #### Further Questions: 1. What are the implications of a matrix commuting with another? 2. How do idempotent matrices behave under matrix powers? 3. Can we generalize the solution for other matrices that satisfy similar properties? 4. How would you solve similar problems for $$3 \times 3$$ matrices? 5. What happens if we modify the entries in $$A$$ slightly? Does $$A^{50}$$ change? --- #### Tip: When working with matrix equations, it’s helpful to check for idempotent properties or symmetries, as these can simplify high powers significantly.

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Math Problem Analysis

Mathematical Concepts

Matrix Multiplication
Commutative Property of Matrices
Matrix Powers
Idempotent Matrices
Linear Algebra

Formulas

Matrix commutation condition AB = BA
Matrix identity: A^2 - 2A + I = 0
Powers of idempotent matrices

Theorems

Matrix Commutation
Idempotent Matrix Theorem

Suitable Grade Level

College

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