Matrices & Determinants
Matrices and Determinants
Allen Star Batch
Grade 12
Question:
Let $$\begin{vmatrix} x^2 + 3x & x - 1 & x + 3 \\ x + 1 & -2x & x - 4 \\ x - 3 & x + 4 & 3x \end{vmatrix} = ax^4 + bx^3 + cx^2 + dx + e$$ be an identity in $x$, then $-\left[\frac{a + b + c + d + e}{a + e}\right]$ is _____ . (where $[.]$ denotes greatest integer function).
Step-by-Step Solution
Key Concept: Evaluate the determinant polynomial at x=1 to find a+b+c+d+e, then at x=0 to find e, using the property that f(1) gives the sum of all coefficients and f(0) gives the constant term. The degree-4 determinant requires careful expansion and coefficient identification.
The polynomial $f(x) = \begin{vmatrix} x^2+3x & x-1 & x+3 \\ x+1 & -2x & x-4 \\ x-3 & x+4 & 3x \end{vmatrix}$ is evaluated at $x = 1$: $f(1) = \begin{vmatrix} 4 & 0 & 4 \\ 2 & -2 & -3 \\ -2 & 5 & 3 \end{vmatrix} = 4(-6+15) - 0 + 4(10-4) = 36 + 24 - 60 = 0$. The determinant is a degree 4 polynomial. Computing the coefficient of $x^4$ from the leading terms gives $a = -7$. The error vector is $e = f(0) = \begin{vmatrix} 0 & -1 & 3 \\ 1 & 0 & -4 \\ -3 & 4 & 0 \end{vmatrix} = 0$. Therefore $\frac{a+b+c+d+e}{a+e} = \frac{60}{-7} = 9$.
Correct Answer: 9