Math Problem Statement

Find the matrix expression 4A + 6B where A = [[9, 5], [5, 2]] and B = [[6, -3], [-2, 6]].

Solution

Let's solve the given problem.

We are given the matrices:

A=[9552],B=[6326],A = \begin{bmatrix} 9 & 5 \\ 5 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 6 & -3 \\ -2 & 6 \end{bmatrix}, and we are tasked with finding 4A+6B4A + 6B.

Step 1: Multiply each matrix by its scalar.

  1. Multiply AA by 4: 4A=4[9552]=[3620208].4A = 4 \cdot \begin{bmatrix} 9 & 5 \\ 5 & 2 \end{bmatrix} = \begin{bmatrix} 36 & 20 \\ 20 & 8 \end{bmatrix}.

  2. Multiply BB by 6: 6B=6[6326]=[36181236].6B = 6 \cdot \begin{bmatrix} 6 & -3 \\ -2 & 6 \end{bmatrix} = \begin{bmatrix} 36 & -18 \\ -12 & 36 \end{bmatrix}.

Step 2: Add the results of 4A4A and 6B6B.

4A+6B=[3620208]+[36181236]=[36+36201820128+36].4A + 6B = \begin{bmatrix} 36 & 20 \\ 20 & 8 \end{bmatrix} + \begin{bmatrix} 36 & -18 \\ -12 & 36 \end{bmatrix} = \begin{bmatrix} 36 + 36 & 20 - 18 \\ 20 - 12 & 8 + 36 \end{bmatrix}.

Step 3: Perform the element-wise addition.

4A+6B=[722844].4A + 6B = \begin{bmatrix} 72 & 2 \\ 8 & 44 \end{bmatrix}.

Final Answer:

4A+6B=[722844].4A + 6B = \begin{bmatrix} 72 & 2 \\ 8 & 44 \end{bmatrix}.

Let me know if you would like a step-by-step breakdown of matrix operations!


Here are 5 related questions for further exploration:

  1. What is the determinant of the resulting matrix 4A+6B4A + 6B?
  2. How would the calculation change if AA and BB were 3x3 matrices?
  3. Can you find ABA - B and then scale it by a factor of 5?
  4. What are the eigenvalues of AA and BB individually?
  5. If AA and BB represented transformations, what is the combined transformation represented by 4A+6B4A + 6B?

Tip: When adding or multiplying matrices, ensure their dimensions match for the operation to be valid!

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Math Problem Analysis

Mathematical Concepts

Matrix Algebra
Matrix Scalar Multiplication
Matrix Addition

Formulas

4A = 4 * A
6B = 6 * B
A + B = A[i][j] + B[i][j]

Theorems

Matrix Addition
Scalar Multiplication of Matrices

Suitable Grade Level

Grades 9-12