Matrices & Determinants
Determinant expansion
Grade 12
Question:
<p>If px^4 + qx^3 + rx^2 + sx + t = \[ \begin{vmatrix} \(x^2+3x\) & \(x-1\) & \(x+3\) \\ \(x+1\) & \(2-x\) & \(x-3\) \\ \(x-3\) & \(x+4\) & 3x \end{vmatrix} \], then t is equal to:</p>
Step-by-Step Solution
Key Concept: Constant term of a polynomial f(x) is f(0). Avoid expansion—substitute directly.
1. **Substitute $x=0$**: Since t = f(0)
2. **Form determinant**: $$ \begin{vmatrix} 0 & -1 & 3 \\ 1 & 2 & -3 \\ -3 & 4 & 0 \end{vmatrix} $$
3. **Expand determinant**: Compute using first row
4. **Simplify**: t = -9 + 30 = 21
Correct Answer: 3