First slide

Introduction to Determinants

Question

If $|\begin{array}{ccc} x^{2} + x & x + 1 & x - 2 \\ 2 x^{2} + 3 x - 1 & 3 x & 3 x - 3 \\ x^{2} + 2 x + 3 & 2 x - 1 & 2 x - 1 \end{array}| = A x + B$ then $A$ is equal to:

Moderate

A

12
B

18
C

24
D

30

Solution

Applying $R_{2} \to R_{2} - R_{1} - R_{3}$ we get,
$A x + B = |\begin{array}{ccc} x^{2} + x & x + 1 & x - 2 \\ - 4 & 0 & 0 \\ x^{2} + 2 x + 3 & 2 x - 1 & 2 x - 1 \end{array}|$
Expanding along $R_{2}$ , we get
$\begin{array}{l} A x + B = (- 1) (- 4) |\begin{array}{cc} x + 1 & x - 2 \\ 2 x - 1 & 2 x - 1 \end{array}| \\ = 4 |\begin{array}{cc} 3 & x - 2 \\ 0 & 2 x - 1 \end{array}| \\ = 4 (3) (2 x - 1) = 24 x - 12 \end{array}$
Thus $A = 24$

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