Math Problem Statement

If \( A^T = \begin{pmatrix} 1 & 2 & 2x+1 \\ 1 & 1 & 2 \\ 0 & -3 & 1 \end{pmatrix}, \) \( B^T = \begin{pmatrix} 1 & 2 & 2 \\ -2 & 1 & 0 \\ 1 & 3 & 4 \end{pmatrix}, \) and \( (A + B)^T = \begin{pmatrix} 2 & 4 & 6-x \\ -1 & 4 & 2 \\ 1 & 0 & 3y-1 \end{pmatrix}, \) find the values of \( x \) and \( y \).

Solution

To solve the given problem, we need to find the values of xx and yy based on the equations provided. Here's the detailed solution:

Step 1: Information given

  • AT=(122x+11120−31)A^T = \begin{pmatrix} 1 & 2 & 2x+1 \\ 1 & 1 & 2 \\ 0 & -3 & 1 \end{pmatrix}
  • BT=(122−210134)B^T = \begin{pmatrix} 1 & 2 & 2 \\ -2 & 1 & 0 \\ 1 & 3 & 4 \end{pmatrix}
  • (A+B)T=(246−x−142103y−1)(A + B)^T = \begin{pmatrix} 2 & 4 & 6-x \\ -1 & 4 & 2 \\ 1 & 0 & 3y-1 \end{pmatrix}

Step 2: Compute (A+B)T(A + B)^T

The transpose of the sum of two matrices is the sum of their transposes. Hence: (A+B)T=AT+BT(A + B)^T = A^T + B^T

Adding ATA^T and BTB^T: AT+BT=(122x+11120−31)+(122−210134)A^T + B^T = \begin{pmatrix} 1 & 2 & 2x+1 \\ 1 & 1 & 2 \\ 0 & -3 & 1 \end{pmatrix} + \begin{pmatrix} 1 & 2 & 2 \\ -2 & 1 & 0 \\ 1 & 3 & 4 \end{pmatrix}

Perform element-wise addition:

= \begin{pmatrix} 2 & 4 & 2x+3 \\ -1 & 2 & 2 \\ 1 & 0 & 5 \end{pmatrix}$$ ### Step 3: Equate with the given $$(A + B)^T$$ The given matrix for $$(A + B)^T$$ is: $$(A + B)^T = \begin{pmatrix} 2 & 4 & 6-x \\ -1 & 4 & 2 \\ 1 & 0 & 3y-1 \end{pmatrix}$$ Comparing elements of the two matrices: 1. From the $$(1, 3)$$ element: $$2x + 3 = 6 - x$$ 2. From the $$(2, 2)$$ element: $$2 = 4$$ (This matches, so no new information.) 3. From the $$(3, 3)$$ element: $$5 = 3y - 1$$ ### Step 4: Solve for $$x$$ From $$2x + 3 = 6 - x$$: $$2x + x = 6 - 3 \implies 3x = 3 \implies x = 1$$ ### Step 5: Solve for $$y$$ From $$5 = 3y - 1$$: $$5 + 1 = 3y \implies 6 = 3y \implies y = 2$$ ### Final Answer: $$x = 1, \quad y = 2$$ Would you like a detailed explanation of how the matrix addition works or any further clarification? --- ### Additional Questions: 1. What is the transpose of a matrix, and how does it affect matrix addition? 2. Can this problem be solved if one element in $$(A + B)^T$$ is missing? Why or why not? 3. How do you compute the determinant of $$A^T + B^T$$? 4. What are the properties of the transpose operation in relation to matrix multiplication? 5. How do you verify if two matrices are equal? ### Tip: Always double-check your arithmetic when solving matrix problems, as small errors can lead to incorrect results!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Matrix Algebra
Matrix Transposition
Matrix Addition
Equation Solving

Formulas

Transpose of a matrix: \((A + B)^T = A^T + B^T\)
Element-wise matrix addition: \( C[i,j] = A[i,j] + B[i,j] \)
Linear equations for solving \( x \) and \( y \)

Theorems

Properties of matrix transposition
Equivalence of matrices

Suitable Grade Level

Grade 11-12

Related Recommendation

Solve for x and y in Matrix Equation A = B^T

Find Values of x and y for A = B Transpose Matrix Equality Problem

Matrix Transposition and Solving for a and b in Matrix Equations

Solving x + y from Matrix Equation A + B = C

Matrix Problem: Find the Value of 2x + y Given At + B = C