Math Problem Statement

The queries provided in the image including matrix operations, complex number matrices, and linear combinations of matrices.

Solution

Let's go through each of the questions:

11. Matrix Properties

Question: Let AA and BB be square matrices such that AB=0AB = 0. Show that if AA is invertible, then B=CB = C.

  • Since AA is invertible, multiply both sides of AB=0AB = 0 by A1A^{-1}: A1AB=A10    B=0.A^{-1}AB = A^{-1}0 \implies B = 0.
  • Therefore, B=0B = 0, which confirms BB is the zero matrix.

12. Complex Matrices

Given Matrices: A=[i00i]andB=[0ii0].A = \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}.

a) Finding A2A^2, A3A^3, and A4A^4:

  1. A2=A×A=[i00i]×[i00i]=[1001]=IA^2 = A \times A = \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} \times \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} = -I.
  2. A3=A2×A=(I)×[i00i]=[i00i]A^3 = A^2 \times A = (-I) \times \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} = \begin{bmatrix} -i & 0 \\ 0 & -i \end{bmatrix}.
  3. A4=A3×A=[i00i]×[i00i]=[1001]=IA^4 = A^3 \times A = \begin{bmatrix} -i & 0 \\ 0 & -i \end{bmatrix} \times \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I.

b) Finding B2B^2: B2=B×B=[0ii0]×[0ii0]=[1001]=I.B^2 = B \times B = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} \times \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} = -I.

13. Guitar Production Increase

Given matrix: A=[7050253510070]A = \begin{bmatrix} 70 & 50 & 25 \\ 35 & 100 & 70 \end{bmatrix} If production is increased by 20%, the new matrix becomes: Anew=1.2×A=[1.2×701.2×501.2×251.2×351.2×1001.2×70]=[8460304212084].A_{\text{new}} = 1.2 \times A = \begin{bmatrix} 1.2 \times 70 & 1.2 \times 50 & 1.2 \times 25 \\ 1.2 \times 35 & 1.2 \times 100 & 1.2 \times 70 \end{bmatrix} = \begin{bmatrix} 84 & 60 & 30 \\ 42 & 120 & 84 \end{bmatrix}.

14. Calorie Burn Matrix

Matrix AA: A=[1091361271596479]A = \begin{bmatrix} 109 & 136 \\ 127 & 159 \\ 64 & 79 \end{bmatrix}

  • A 120-pound person and a 150-pound person bicycled for 40 minutes, jogged for 10 minutes, and walked for 60 minutes. To organize the amounts of time spent exercising in matrix BB:

B=[404010106060].B = \begin{bmatrix} 40 & 40 \\ 10 & 10 \\ 60 & 60 \end{bmatrix}.

15. Linear Combination of Matrices

Given matrices: X=[1201],Y=[1032],Z=[3412],W=[3241]X = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}, \quad Y = \begin{bmatrix} -1 & 0 \\ 3 & 2 \end{bmatrix}, \quad Z = \begin{bmatrix} 3 & 4 \\ -1 & 2 \end{bmatrix}, \quad W = \begin{bmatrix} 3 & 2 \\ -4 & -1 \end{bmatrix}

a) Find scalars a,b,ca, b, c such that W=aX+bY+cZW = aX + bY + cZ.

To find a,b,a, b, and cc, set up the equation: W=aX+bY+cZ.W = aX + bY + cZ. Equate corresponding elements of WW and solve the resulting linear system.

b) If aX+bY+cZ=0aX + bY + cZ = 0, show that a=b=c=0a = b = c = 0.

This problem can be solved by checking if the determinant of the matrix formed by X,Y,X, Y, and ZZ (treated as vectors) is non-zero.

Let me know if you'd like a detailed explanation or solution for any of the parts above, and feel free to ask additional questions.

Further Questions

  1. How do you prove invertibility for matrices in general?
  2. What are the properties of matrix multiplication that apply to the identity and zero matrices?
  3. How can you determine the power of a complex matrix like AA in question 12?
  4. What does it mean for a matrix to be similar or share properties with another matrix?
  5. How can production increases be modeled using matrix transformations?

Tip: When solving matrix equations, always verify the determinant and properties of the matrices involved to simplify your calculations.

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Math Problem Analysis

Mathematical Concepts

Matrix Algebra
Matrix Inversion
Matrix Multiplication
Linear Algebra
Complex Numbers
Scalar Multiplication

Formulas

Matrix multiplication: AB = 0
Matrix inversion: A^-1
Matrix power: A^2, A^3, A^4
Scalar matrix equation: W = aX + bY + cZ

Theorems

Matrix Inversion Theorem
Properties of Identity and Zero Matrices

Suitable Grade Level

College-Level Linear Algebra