Math Problem Statement
Determine whether B is the inverse of A. A =
2 1 −1
5 3 −5 −1 −5 2 , B =
5 −1 5
−3 4 5 −3 4 5 AB =
Is B the inverse of A?
Yes No
Solution
To determine if matrix is the inverse of matrix , we must check if the product , where is the identity matrix.
Let's begin by calculating the matrix multiplication of and .
Given:
2 & 1 & -1 \\ 5 & 3 & -5 \\ -1 & -5 & 2 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 5 & -1 & 5 \\ -3 & 4 & 5 \\ -3 & 4 & 5 \end{pmatrix}$$ The multiplication $$AB$$ is carried out as follows: $$AB = \begin{pmatrix} (2 \times 5 + 1 \times -3 + -1 \times -3) & (2 \times -1 + 1 \times 4 + -1 \times 4) & (2 \times 5 + 1 \times 5 + -1 \times 5) \\ (5 \times 5 + 3 \times -3 + -5 \times -3) & (5 \times -1 + 3 \times 4 + -5 \times 4) & (5 \times 5 + 3 \times 5 + -5 \times 5) \\ (-1 \times 5 + -5 \times -3 + 2 \times -3) & (-1 \times -1 + -5 \times 4 + 2 \times 4) & (-1 \times 5 + -5 \times 5 + 2 \times 5) \end{pmatrix}$$ Simplifying each entry: First row: $$(10 + -3 + 3) = 10, \quad (-2 + 4 + -4) = -2, \quad (10 + 5 + -5) = 10$$ Second row: $$(25 + -9 + 15) = 31, \quad (-5 + 12 + -20) = -13, \quad (25 + 15 + -25) = 15$$ Third row: $$(-5 + 15 + -6) = 4, \quad (1 + -20 + 8) = -11, \quad (-5 + -25 + 10) = -20$$ Thus, the product $$AB$$ is: $$AB = \begin{pmatrix} 10 & -2 & 10 \\ 31 & -13 & 15 \\ 4 & -11 & -20 \end{pmatrix}$$ Since $$AB$$ is not the identity matrix (which would be $$I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$), matrix $$B$$ is **not** the inverse of matrix $$A$$. ### Conclusion: **No**, $$B$$ is not the inverse of $$A$$. Do you want any further details or explanations? Here are 5 related questions for deeper exploration: 1. What is the general process for finding the inverse of a 3x3 matrix? 2. Can you compute the inverse of matrix $$A$$ manually? 3. What are the properties of an identity matrix? 4. How would you determine if a matrix is singular? 5. How do you check if a matrix multiplication yields the identity matrix? **Tip:** For two matrices $$A$$ and $$B$$ to be inverses, their product must equal the identity matrix both as $$AB$$ and $$BA$$.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Multiplication
Matrix Inverses
Formulas
Matrix multiplication: (AB)ij = Σ Aik Bkj
Inverse matrix condition: AB = I
Theorems
Properties of Matrix Inverses
Identity Matrix Theorem
Suitable Grade Level
Grades 11-12, College Level