Math Problem Statement

Please help me solve this question

Find the matrix C given BC and the inverse of matrix B.

Solution

The problem asks to find matrix CC given the following information:

  1. BC=[123456789]BC = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix},
  2. B−1=[123014001]B^{-1} = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{bmatrix}.

Step-by-Step Solution:

To find matrix CC, we use the equation:

BC=ABC = A where A=[123456789]A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}.

To isolate CC, we multiply both sides of the equation by B−1B^{-1} from the left:

B−1BC=B−1AB^{-1} BC = B^{-1} A

Since B−1B=IB^{-1}B = I (the identity matrix), we are left with:

C=B−1AC = B^{-1} A

Now, let's compute CC by multiplying B−1B^{-1} with AA:

A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$$ Multiplying these two matrices: $$C = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$$ ### Matrix multiplication: 1. First row of $$B^{-1}$$ with each column of $$A$$: - $$(1 \cdot 1) + (2 \cdot 4) + (3 \cdot 7) = 1 + 8 + 21 = 30$$ - $$(1 \cdot 2) + (2 \cdot 5) + (3 \cdot 8) = 2 + 10 + 24 = 36$$ - $$(1 \cdot 3) + (2 \cdot 6) + (3 \cdot 9) = 3 + 12 + 27 = 42$$ 2. Second row of $$B^{-1}$$ with each column of $$A$$: - $$(0 \cdot 1) + (1 \cdot 4) + (4 \cdot 7) = 0 + 4 + 28 = 32$$ - $$(0 \cdot 2) + (1 \cdot 5) + (4 \cdot 8) = 0 + 5 + 32 = 37$$ - $$(0 \cdot 3) + (1 \cdot 6) + (4 \cdot 9) = 0 + 6 + 36 = 42$$ 3. Third row of $$B^{-1}$$ with each column of $$A$$: - $$(0 \cdot 1) + (0 \cdot 4) + (1 \cdot 7) = 0 + 0 + 7 = 7$$ - $$(0 \cdot 2) + (0 \cdot 5) + (1 \cdot 8) = 0 + 0 + 8 = 8$$ - $$(0 \cdot 3) + (0 \cdot 6) + (1 \cdot 9) = 0 + 0 + 9 = 9$$ So, the matrix $$C$$ is: $$C = \begin{bmatrix} 30 & 36 & 42 \\ 32 & 37 & 42 \\ 7 & 8 & 9 \end{bmatrix}$$ ### Final Answer: $$C = \begin{bmatrix} 30 & 36 & 42 \\ 32 & 37 & 42 \\ 7 & 8 & 9 \end{bmatrix}$$ Would you like further details on the steps? Here are 5 follow-up questions: 1. What are the properties of an invertible matrix? 2. How do you calculate the inverse of a 3x3 matrix manually? 3. Why does multiplying a matrix by its inverse yield the identity matrix? 4. What are the applications of matrix inversion in solving linear systems? 5. How can we verify if the multiplication between two matrices is correct? **Tip:** Always double-check your matrix multiplication to avoid simple errors in calculations.

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Math Problem Analysis

Mathematical Concepts

Matrix Multiplication
Inverse of a Matrix
Linear Algebra

Formulas

C = B^{-1} A
Matrix multiplication: (AB)_{ij} = \sum_{k} A_{ik}B_{kj}

Theorems

Inverse Matrix Theorem: B^{-1} B = I (Identity Matrix)

Suitable Grade Level

Undergraduate Math or Linear Algebra Course

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