Math Problem Statement
Given matrices, find specific elements, compute expressions, determine transposes, perform matrix addition, and subtraction.
Solution
Let's break down the problems step by step from the image you uploaded:
1. Matrix :
1 & 0 & -2 \\ 3 & 1 & 4 \\ -5 & 8 & 7 \end{pmatrix}$$ You are asked to find the following elements from matrix $$A$$: - $$a_{32}$$ (the element in the 3rd row, 2nd column), - $$a_{21}$$ (the element in the 2nd row, 1st column), - $$a_{13}$$ (the element in the 1st row, 3rd column), - $$a_{22}$$ (the element in the 2nd row, 2nd column). From matrix $$A$$, we can directly locate: - $$a_{32} = 8$$, - $$a_{21} = 3$$, - $$a_{13} = -2$$, - $$a_{22} = 1$$. ### 2. Matrix $$B$$: $$B = \begin{pmatrix} -3 & 5 & -2 \\ 3 & 0 & 4 \\ 1 & 4 & 3 \end{pmatrix}$$ You are asked to find: $$b_{11}^2 + 3 \cdot b_{13} - b_{32}$$ From matrix $$B$$: - $$b_{11} = -3$$, - $$b_{13} = -2$$, - $$b_{32} = 1$$. Now we can compute: $$b_{11}^2 + 3 \cdot b_{13} - b_{32} = (-3)^2 + 3 \cdot (-2) - 1 = 9 - 6 - 1 = 2.$$ ### 3. Transpositions of Matrices $$C$$, $$E$$, and $$D$$: The transpose of a matrix is obtained by switching its rows and columns. Matrix $$C$$: $$C = \begin{pmatrix} 2 & 4 \\ -3 & 0 \\ 1 & 2 \end{pmatrix}$$ Transpose of $$C$$ (denoted as $$C^T$$): $$C^T = \begin{pmatrix} 2 & -3 & 1 \\ 4 & 0 & 2 \end{pmatrix}$$ Matrix $$E$$: $$E = \begin{pmatrix} 1 & 3 & -4 \\ -2 & 4 & 5 \end{pmatrix}$$ Transpose of $$E$$ (denoted as $$E^T$$): $$E^T = \begin{pmatrix} 1 & -2 \\ 3 & 4 \\ -4 & 5 \end{pmatrix}$$ Matrix $$D$$: $$D = \begin{pmatrix} -1 & 0 & 2 & 7 \\ 3 & 4 & 1 & 5 \\ -2 & 1 & 1 & -3 \end{pmatrix}$$ Transpose of $$D$$ (denoted as $$D^T$$): $$D^T = \begin{pmatrix} -1 & 3 & -2 \\ 0 & 4 & 1 \\ 2 & 1 & 1 \\ 7 & 5 & -3 \end{pmatrix}$$ ### 4. Sum of Matrices $$A$$ and $$B$$: Matrix $$A$$ is: $$A = \begin{pmatrix} 1 & -2 \\ 3 & 4 \end{pmatrix}$$ Matrix $$B$$ is: $$B = \begin{pmatrix} 3 & -7 \\ -6 & 0 \end{pmatrix}$$ The sum of these matrices is calculated by adding corresponding elements: $$A + B = \begin{pmatrix} 1 + 3 & -2 + (-7) \\ 3 + (-6) & 4 + 0 \end{pmatrix} = \begin{pmatrix} 4 & -9 \\ -3 & 4 \end{pmatrix}$$ ### 5. Difference of Matrices $$C$$ and $$D$$: Matrix $$C$$ is: $$C = \begin{pmatrix} 2 & 0 & 1 \\ -1 & 3 & 0 \end{pmatrix}$$ Matrix $$D$$ is: $$D = \begin{pmatrix} -1 & 3 & 4 \\ -3 & -5 & 7 \\ -2 & 0 & 2 \end{pmatrix}$$ The difference of matrices $$C - D$$ is calculated by subtracting corresponding elements: $$C - D = \begin{pmatrix} 2 - (-1) & 0 - 3 & 1 - 4 \\ -1 - (-3) & 3 - (-5) & 0 - 7 \end{pmatrix} = \begin{pmatrix} 3 & -3 & -3 \\ 2 & 8 & -7 \end{pmatrix}$$ Let me know if you'd like further explanations on any of these steps or additional questions. Here are some related questions you may find interesting: 1. What are the properties of matrix transposition? 2. How do you find the determinant of a 3x3 matrix? 3. How do you perform matrix multiplication? 4. What is the inverse of a matrix and how is it calculated? 5. How do you solve systems of linear equations using matrices? **Tip:** Always double-check matrix dimensions before performing operations like addition, subtraction, or multiplication, as they must be compatible.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Matrix Transpose
Matrix Addition and Subtraction
Matrix Element Extraction
Basic Arithmetic Operations
Formulas
Matrix Transpose: (A^T)ij = Aji
Matrix Addition: (A + B)ij = Aij + Bij
Matrix Subtraction: (A - B)ij = Aij - Bij
Theorems
Basic properties of matrix operations
Suitable Grade Level
Grades 10-12
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