Question:
If
\[
P = \begin{bmatrix} 1 & 0 & 1 \\ 2 & 0 & 1 \\ 0 & 0 & -1 \end{bmatrix} \quad \text{and} \quad 6P^{-1} = aI_3 + bP - P^2, \quad \text{then the ordered pair} \quad (a,b) \quad \text{is}
\]
If
\[ P = \begin{bmatrix} 1 & 0 & 1 \\ 2 & 0 & 1 \\ 0 & 0 & -1 \end{bmatrix} \quad \text{and} \quad 6P^{-1} = aI_3 + bP - P^2, \quad \text{then the ordered pair} \quad (a,b) \quad \text{is} \]
Show Hint
To solve matrix equations, find the inverse of the matrix and substitute into the given equation to solve for unknowns.
Updated On: Dec 12, 2025
- (3,2)
- (2,3)
- (4,5)
- (5,4)
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The Correct Option is A
Solution and Explanation
Step 1: Find the inverse of \( P \).
To find \( P^{-1} \), we use the formula for the inverse of a 3x3 matrix. After calculating, we find that
Substitute \( P^{-1} \) into the given equation \( 6P^{-1} = aI_3 + bP - P^2 \). By solving for \( a \) and \( b \), we obtain the values \( a = 4 \) and \( b = 5 \).
Step 3: Conclusion.
The correct answer is (C) (4,5).
To find \( P^{-1} \), we use the formula for the inverse of a 3x3 matrix. After calculating, we find that
\[ P^{-1} = \begin{bmatrix} 1 & 0 & 1 \\ 2 & 0 & 1 \\ 0 & 0 & -1 \end{bmatrix} \]
Step 2: Substitute into the equation.Substitute \( P^{-1} \) into the given equation \( 6P^{-1} = aI_3 + bP - P^2 \). By solving for \( a \) and \( b \), we obtain the values \( a = 4 \) and \( b = 5 \).
Step 3: Conclusion.
The correct answer is (C) (4,5).
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