Given that Q = \(\begin{pmatrix} 6 & 0 \\ 4 & 5 \end{pmatrix}\) and Q + P = \(\begin{pmatrix} 7 & -2 \\ 6 & 8 \end{pmatrix}\). Evaluate | Q + 2P|.
A
90
B
96
C
102
D
120
Verified Correct Answer: D.
Step-by-Step Mathematical Solution
Q + 2P = Q + 2(Q + P - Q) = 2(Q + P) - Q
Q + P = \(\begin{pmatrix} 7 & -2 \\ 6 & 8 \end{pmatrix}\)
2(Q + P) = \(\begin{pmatrix} 14 & -4 \\ 12 & 16 \end{pmatrix}\)
Q + 2P = 2(Q + P) - Q = \(\begin{pmatrix} 14-6 & -4-0 \\ 12-4 & 16-5 \end{pmatrix} = \begin{pmatrix} 8 & -4 \\ 8 & 11 \end{pmatrix}\)
Determinant: |Q + 2P| = 8 \(\times\) 11 - (-4)\(\times\) 8 = 88 + 32 = 120.
Source: Myschool Past Questions Archive (Record #10416)