Matrix Multiplication with Diagonal Matrices: A · D and D

Math Problem Statement

Calculate the matrix products A · D and D · A for matrix A and diagonal matrix D.

Solution

Let's solve this step by step.

Given:

Matrix $A$ : $-3 & 1 & -2 \\ 2 & -3 & 3 \\ -1 & 2 & 3 \end{pmatrix}$$$
Diagonal matrix $D$ : $-2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 3 \end{pmatrix}$$$

We are tasked to calculate $A \cdot D$ and $D \cdot A$ .

Calculation of $A \cdot D$ :

To compute $A \cdot D$ , each row of $A$ is multiplied by the corresponding diagonal entry in $D$ . This simplifies to scaling the columns of $A$ by the diagonal entries of $D$ .

-3 & 1 & -2 \\ 2 & -3 & 3 \\ -1 & 2 & 3 \end{pmatrix} \cdot \begin{pmatrix} -2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 3 \end{pmatrix}$$ Result: $$A \cdot D = \begin{pmatrix} (-3)(-2) & (1)(0) & (-2)(3) \\ (2)(-2) & (-3)(0) & (3)(3) \\ (-1)(-2) & (2)(0) & (3)(3) \end{pmatrix}$$ Simplify: $$A \cdot D = \begin{pmatrix} 6 & 0 & -6 \\ -4 & 0 & 9 \\ 2 & 0 & 9 \end{pmatrix}$$ --- ### Calculation of $$D \cdot A$$: To compute $$D \cdot A$$, each column of $$A$$ is scaled by the corresponding diagonal entry in $$D$$. $$D \cdot A = \begin{pmatrix} -2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 3 \end{pmatrix} \cdot \begin{pmatrix} -3 & 1 & -2 \\ 2 & -3 & 3 \\ -1 & 2 & 3 \end{pmatrix}$$ Result: $$D \cdot A = \begin{pmatrix} (-2)(-3) & (-2)(1) & (-2)(-2) \\ (0)(2) & (0)(-3) & (0)(3) \\ (3)(-1) & (3)(2) & (3)(3) \end{pmatrix}$$ Simplify: $$D \cdot A = \begin{pmatrix} 6 & -2 & 4 \\ 0 & 0 & 0 \\ -3 & 6 & 9 \end{pmatrix}$$ --- ### Final Results: 1. $$A \cdot D = \begin{pmatrix} 6 & 0 & -6 \\ -4 & 0 & 9 \\ 2 & 0 & 9 \end{pmatrix}$$ 2. $$D \cdot A = \begin{pmatrix} 6 & -2 & 4 \\ 0 & 0 & 0 \\ -3 & 6 & 9 \end{pmatrix}$$ Would you like further details on the calculations or have any specific questions? --- #### 5 Related Questions: 1. How do diagonal matrices affect matrix multiplication in general? 2. What are the properties of diagonal matrices in multiplication? 3. How can you compute the determinant of a product of matrices like $$A \cdot D$$? 4. What is the computational complexity of matrix multiplication for general matrices versus diagonal matrices? 5. What is the geometric interpretation of scaling rows or columns in matrix multiplication? #### Tip: When multiplying by a diagonal matrix, focus on scaling the rows or columns, making the computation much simpler!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Matrix Multiplication
Diagonal Matrices
Linear Algebra

Formulas

Matrix multiplication formula: (AB)_{ij} = ∑_k A_{ik}B_{kj}

Theorems

Diagonal matrices simplify matrix multiplication by scaling rows or columns.

Suitable Grade Level

Grades 11-12 or College Level

Related Recommendation

Matrix Multiplication: Calculating AB for Given Matrices

Matrix Multiplication and Commutativity Problem: Compute AD and DA, Find Matrix B

Commutativity of Matrix Multiplication: Diagonal Matrix A and Arbitrary Matrix B

Matrix Operations with Sparse and Diagonal Matrices: Solution to (C + D)a

Matrix Addition: A Detailed Explanation with Example