Math Problem Statement
Consider this math question: Let matrix $A = ||begin{bmatrix} 3 & -2 & 1 111085& 41111-6&28-111end{bmatrix}$. Define matrix $B$ as the transpose of the transpose of the transpose of matrix $A^T$. Using row reduction, find the determinant of matrix $A + B$."
Solution
To solve this problem, we need to go through a few steps to understand the given matrices and the operations involved. Let's break it down:
Step 1: Understanding Matrix
You provided the matrix as:
3 & -2 & 1 \\ 1 & 1 & 0 \\ 8 & 5 & 4 \\ -1 & 1 & -6 \\ & 2 & 8 & -1 \end{bmatrix}$$ This matrix is a 4x3 matrix, meaning it has 4 rows and 3 columns. ### Step 2: Understanding the Matrix $$B$$ We are given that matrix $$B$$ is defined as the transpose of the transpose of the transpose of matrix $$A^T$$. Let’s break this down: - The transpose of matrix $$A$$, written as $$A^T$$, changes the rows of $$A$$ into columns. - If we take the transpose of $$A^T$$ (which is $$A^{T T}$$), we are back to the original matrix $$A$$ itself. - Finally, taking the transpose again (i.e., $$A^{T T T}$$) means we are back to matrix $$A^T$$. Thus, matrix $$B$$ is simply the transpose of matrix $$A$$, i.e., $$B = A^T$$. ### Step 3: Finding Matrix $$A + B$$ Now, we need to find the sum of $$A + B$$. Since $$B = A^T$$, we compute $$A + A^T$$. To do this, let's first compute the transpose of matrix $$A$$, which switches rows and columns. #### Matrix $$A^T$$: $$A^T = \begin{bmatrix} 3 & 1 & 8 & -1 \\ -2 & 1 & 5 & 1 \\ 1 & 0 & 4 & -6 \end{bmatrix}$$ #### Adding $$A + A^T$$: Now, we add $$A$$ and $$A^T$$ element-wise: $$A + A^T = \begin{bmatrix} 3 & -2 & 1 \\ 1 & 1 & 0 \\ 8 & 5 & 4 \\ -1 & 1 & -6 \end{bmatrix} + \begin{bmatrix} 3 & 1 & 8 & -1 \\ -2 & 1 & 5 & 1 \\ 1 & 0 & 4 & -6 \end{bmatrix}$$ However, there is an issue in matrix dimensions, as $$A$$ is a 4x3 matrix, and $$A^T$$ is a 3x4 matrix. The matrices cannot be added directly since they do not have the same dimensions. ### Conclusion: It seems that there is a misunderstanding in the matrix dimensions of $$A$$ and $$B$$. Matrix $$A$$ is a 4x3 matrix, while matrix $$B = A^T$$ is a 3x4 matrix. Because of this dimensional mismatch, you cannot compute $$A + B$$ as defined. If you meant to provide square matrices for $$A$$, the determinant would be feasible, but as written, this problem does not have a solution for the determinant of $$A + B$$ due to the size inconsistency. Let me know if you'd like further clarification on this issue or if you would like to adjust the matrix dimensions!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Transposition
Matrix Addition
Determinants
Row Reduction
Formulas
Transpose of a matrix: (A^T)^T = A
Matrix Addition: (A + B)_{ij} = A_{ij} + B_{ij}
Determinant of a square matrix using row reduction
Theorems
Properties of Transpose: (A^T)^T = A
Determinant Properties in Row Operations
Addition of Matrices Requires Same Dimensions
Suitable Grade Level
Undergraduate Linear Algebra