Question

If \( P, Q \) and \( R \) are \( 3 \times 3 \) matrices such that} \[ P = \begin{bmatrix} 3x^2 + x + 3 & 2x^2 - x + 4 & 7x^2 + 8x + 5 \\ 5x^2 + 3x + 2 & 4x^2 - 2x - 1 & 7x^2 + 5x + 8 \\ 3x^2 + 2x + 5 & 4x^2 - x - 2 & 3x^2 + 8x + 7 \end{bmatrix} = Px^2 + Qx + R \] then det \( R = \) ?

• A. \( 0 \)
• B. \( 136 \)
• C. \( 48 \)
• D. \( -72 \)

Hint:

When given a polynomial matrix like \( Px^2 + Qx + R \), always collect the constant (independent of \( x \)) terms to form matrix \( R \).

Answer 1

The Correct Option is B

Step 1: Identify constant term matrix \( R \).
Matrix \( R \) is the constant part (i.e., terms independent of \( x \)). Extract the constants: \[ R = \begin{bmatrix} 3 & 4 & 5 \\ 2 & -1 & 8 \\ 5 & -2 & 7 \end{bmatrix} \] Step 2: Compute determinant of \( R \) Use cofactor expansion: \[ \det(R) = 3 \begin{vmatrix} -1 & 8
-2 & 7 \end{vmatrix} - 4 \begin{vmatrix} 2 & 8
5 & 7 \end{vmatrix} + 5 \begin{vmatrix} 2 & -1
5 & -2 \end{vmatrix} \] \[ = 3((-1)(7) - (8)(-2)) - 4((2)(7) - (8)(5)) + 5((2)(-2) - (-1)(5))
= 3(-7 + 16) - 4(14 - 40) + 5(-4 + 5)
= 3(9) - 4(-26) + 5(1) = 27 + 104 + 5 = \boxed{136} \]