Math Problem Statement
Examine the product of the two matrices to determine if each is the inverse of the other.
left bracket Start 3 By 3 Matrix 1st Row 1st Column 1 2nd Column negative 1 3rd Column 1 2nd Row 1st Column 0 2nd Column 1 3rd Column negative 1 3rd Row 1st Column 4 2nd Column 1 3rd Column 0 EndMatrix right bracket
1
−1
1
0
1
−1
4
1
0
left bracket Start 3 By 3 Matrix 1st Row 1st Column 1 2nd Column 1 3rd Column 0 2nd Row 1st Column negative 4 2nd Column negative 5 3rd Column 1 3rd Row 1st Column negative 4 2nd Column negative 5 3rd Column 0 EndMatrix right bracket
1
1
0
−4
−5
1
−4
−5
0
Question content area bottom
Part 1
Are the matrices inverses of each other?
Yes
YesYes
No
No
Solution
To determine whether two matrices are inverses of each other, we need to check if their product results in the identity matrix. Specifically, if matrices and are inverses of each other, then and , where is the identity matrix.
Given matrices:
\begin{pmatrix} 1 & -1 & 1 \\ 0 & 1 & -1 \\ 4 & 1 & 0 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 1 & 1 & 0 \\ -4 & -5 & 1 \\ -4 & -5 & 0 \end{pmatrix}$$ Let's compute the product $$A \times B$$: $$A \times B = \begin{pmatrix} 1 & -1 & 1 \\ 0 & 1 & -1 \\ 4 & 1 & 0 \end{pmatrix} \cdot \begin{pmatrix} 1 & 1 & 0 \\ -4 & -5 & 1 \\ -4 & -5 & 0 \end{pmatrix}$$ ### Step-by-Step Multiplication 1. **First row, first column:** $$(1 \times 1) + (-1 \times -4) + (1 \times -4) = 1 + 4 - 4 = 1$$ 2. **First row, second column:** $$(1 \times 1) + (-1 \times -5) + (1 \times -5) = 1 + 5 - 5 = 1$$ 3. **First row, third column:** $$(1 \times 0) + (-1 \times 1) + (1 \times 0) = 0 - 1 + 0 = -1$$ 4. **Second row, first column:** $$(0 \times 1) + (1 \times -4) + (-1 \times -4) = 0 - 4 + 4 = 0$$ 5. **Second row, second column:** $$(0 \times 1) + (1 \times -5) + (-1 \times -5) = 0 - 5 + 5 = 0$$ 6. **Second row, third column:** $$(0 \times 0) + (1 \times 1) + (-1 \times 0) = 0 + 1 + 0 = 1$$ 7. **Third row, first column:** $$(4 \times 1) + (1 \times -4) + (0 \times -4) = 4 - 4 + 0 = 0$$ 8. **Third row, second column:** $$(4 \times 1) + (1 \times -5) + (0 \times -5) = 4 - 5 + 0 = -1$$ 9. **Third row, third column:** $$(4 \times 0) + (1 \times 1) + (0 \times 0) = 0 + 1 + 0 = 1$$ So, the product $$A \times B$$ is: $$A \times B = \begin{pmatrix} 1 & 1 & -1 \\ 0 & 0 & 1 \\ 0 & -1 & 1 \end{pmatrix}$$ This result is not the identity matrix. Since $$A \times B$$ does not equal the identity matrix, matrices $$A$$ and $$B$$ are **not** inverses of each other. ### Final Answer: No, the matrices are not inverses of each other. Do you have any questions or need further details? Here are five related questions you might find interesting: 1. What properties must a matrix satisfy to have an inverse? 2. How can we determine if a matrix is singular or non-singular? 3. Can non-square matrices have inverses? 4. What is the process for finding the inverse of a 3x3 matrix? 5. How does the determinant of a matrix relate to its invertibility? **Tip:** Always check if the product of two matrices results in the identity matrix when verifying if they are inverses.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Operations
Inverse Matrices
Identity Matrix
Formulas
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Theorems
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Suitable Grade Level
Advanced High School