Question

Consider a matrix \( A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & -2 \\ 0 & 1 & 1 \end{bmatrix} \). The matrix \( A \) satisfies the equation \( 6A^{-1} = A^2 + cA + dI \), where \( c \) and \( d \) are scalars and \( I \) is the identity matrix. Then \( (c + d) \) is equal to

• A. 5
• B. 17
• C. -6
• D. 11

Hint:

When solving for constants in matrix equations, compute the matrix inverse and use matrix multiplication to solve for the unknowns.

Answer 1

The Correct Option is A

Step 1: Matrix Inversion and Squaring.
We need to solve for \( c \) and \( d \). First, we compute \( A^{-1} \) and \( A^2 \). The inverse of \( A \) is found using the formula for the inverse of a 3x3 matrix, and \( A^2 \) is computed by matrix multiplication. \[ A^2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & -2 \\ 0 & 1 & 1 \end{bmatrix}^2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 16 & -8 \\ 0 & 5 & -1 \end{bmatrix}. \] Step 2: Substituting into the given equation.
Next, we substitute the expressions for \( A^{-1} \) and \( A^2 \) into the given equation \( 6A^{-1} = A^2 + cA + dI \). Solving this equation gives us the values of \( c \) and \( d \). \[ c + d = 5. \] Step 3: Conclusion.
The correct answer is (A), as \( c + d = 5 \).