Question

A is a $3 \times 3$ matrix satisfying $A^3 - 5A^2 + 7A + I = 0$. If $A^5 - 6A^4 + 12A^3 - 6A^2 + 2A + 2I = lA + mI$, then $l + m =$

• A. 5
• B. -1
• C. 4
• D. 2

Hint:

When a matrix satisfies a polynomial equation, higher powers of the matrix can be reduced by dividing the given polynomial by the characteristic polynomial. The remainder directly gives the simplified form.

Answer 1

The Correct Option is A

Step 1: Understand the given expressions:
We are given the matrix equation \[ P(A) = A^3 - 5A^2 + 7A + I = 0 \] and we need to simplify \[ Q(A) = A^5 - 6A^4 + 12A^3 - 6A^2 + 2A + 2I \]
Step 2: Divide the polynomials:
Consider the corresponding polynomials \[ Q(x) = x^5 - 6x^4 + 12x^3 - 6x^2 + 2x + 2 \] and \[ P(x) = x^3 - 5x^2 + 7x + 1 \] On performing polynomial division of $Q(x)$ by $P(x)$, we obtain \[ Q(x) = (x^2 - x)P(x) + (3x + 2) \] Thus, the quotient is $x^2 - x$ and the remainder is $3x + 2$.

Step 3: Substitute the matrix $A$:
Replacing $x$ by $A$, we get \[ Q(A) = (A^2 - A)P(A) + (3A + 2I) \]
Step 4: Use the given condition:
Since $P(A) = 0$, the first term vanishes, giving \[ Q(A) = 3A + 2I \]
Step 5: Compare and find values:
Given that $Q(A) = lA + mI$, we compare terms to obtain \[ l = 3, m = 2 \]
Step 6: Final result:
\[ l + m = 3 + 2 = 5 \] Final Answer: $l + m = 5$