Math Problem Statement
Perform the following matrix operations:
(i) Transpose the matrix [[5, -6], [4, 9]].
(ii) Compute 2[[x, x + y], [-2, -x]] - 3[[-x, -2y], [x, -1]].
(iii) Multiply the matrices [[6, 0], [1, -1]] and [[-3, -2], [3x, -1]].
Solution
Let's solve the provided matrix operations one by one. I’ll transcribe the problem and then proceed with detailed solutions.
Question (i):
Transpose the matrix
5 & -6 \\ 4 & 9 \end{bmatrix}$$ #### Solution: The transpose of a matrix is obtained by swapping its rows and columns. Transpose the given matrix: $$\text{Transpose} = \begin{bmatrix} 5 & -6 \\ 4 & 9 \end{bmatrix}^T = \begin{bmatrix} 5 & 4 \\ -6 & 9 \end{bmatrix}$$ --- ### Question (ii): Compute the operation: $$2\begin{bmatrix} x & x + y \\ -2 & -x \end{bmatrix} - 3\begin{bmatrix} -x & -2y \\ x & -1 \end{bmatrix}$$ #### Solution: First, compute each matrix product: 1. Multiply the first matrix by 2: $$2\begin{bmatrix} x & x + y \\ -2 & -x \end{bmatrix} = \begin{bmatrix} 2x & 2(x + y) \\ -4 & -2x \end{bmatrix} = \begin{bmatrix} 2x & 2x + 2y \\ -4 & -2x \end{bmatrix}$$ 2. Multiply the second matrix by 3: $$3\begin{bmatrix} -x & -2y \\ x & -1 \end{bmatrix} = \begin{bmatrix} -3x & -6y \\ 3x & -3 \end{bmatrix}$$ Now subtract the second matrix from the first: $$\begin{bmatrix} 2x & 2x + 2y \\ -4 & -2x \end{bmatrix} - \begin{bmatrix} -3x & -6y \\ 3x & -3 \end{bmatrix} = \begin{bmatrix} 2x - (-3x) & 2x + 2y - (-6y) \\ -4 - 3x & -2x - (-3) \end{bmatrix}$$ Simplify: $$= \begin{bmatrix} 2x + 3x & 2x + 2y + 6y \\ -4 - 3x & -2x + 3 \end{bmatrix} = \begin{bmatrix} 5x & 2x + 8y \\ -4 - 3x & -2x + 3 \end{bmatrix}$$ --- ### Question (iii): Multiply the matrices: $$\begin{bmatrix} 6 & 0 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} -3 & -2 \\ 3x & -1 \end{bmatrix}$$ #### Solution: Perform matrix multiplication: $$\begin{bmatrix} 6 & 0 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} -3 & -2 \\ 3x & -1 \end{bmatrix} = \begin{bmatrix} (6)(-3) + (0)(3x) & (6)(-2) + (0)(-1) \\ (1)(-3) + (-1)(3x) & (1)(-2) + (-1)(-1) \end{bmatrix}$$ Simplify each element: $$= \begin{bmatrix} -18 + 0 & -12 + 0 \\ -3 - 3x & -2 + 1 \end{bmatrix} = \begin{bmatrix} -18 & -12 \\ -3 - 3x & -1 \end{bmatrix}$$ --- ### Summary of Results: 1. (i) Transposed matrix: $$\begin{bmatrix} 5 & 4 \\ -6 & 9 \end{bmatrix}$$ 2. (ii) Result: $$\begin{bmatrix} 5x & 2x + 8y \\ -4 - 3x & -2x + 3 \end{bmatrix}$$ 3. (iii) Product: $$\begin{bmatrix} -18 & -12 \\ -3 - 3x & -1 \end{bmatrix}$$ --- Would you like any further details or explanations on these operations? Here are some related questions to explore: 1. What is the determinant of a matrix, and can we compute it for one of these? 2. How do eigenvalues relate to a matrix like in question (i)? 3. What are the properties of matrix multiplication? 4. How can matrix operations be applied in solving systems of linear equations? 5. What happens when multiplying matrices with symbolic variables like $$x$$ or $$y$$? **Tip**: Always verify the dimensions of matrices before performing operations to ensure validity.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Matrix Transposition
Matrix Multiplication
Matrix Addition and Subtraction
Formulas
Matrix Transposition: Swap rows and columns.
Matrix Multiplication: Sum of products of rows and columns.
Matrix Scalar Multiplication: Multiply each element by the scalar.
Theorems
Properties of Matrix Operations
Suitable Grade Level
Grades 10-12