(a) Evaluate: \(\int \limits_1^2 (2x^3 - 4x +…

Question 1 Report

(a) Evaluate: \(\int \limits_1^2 (2x^3 - 4x + 3) dx\)

(b) Given that P\(^{-1}\) = \(\begin{pmatrix} -1 & 1 \\ 4 & -3 \end{pmatrix}\), find the matrix P.

Answer

Answer Details

(a)

\[\int_1^2 (2x^3 - 4x + 3)\,dx = \left[\frac{x^4}{2} - 2x^2 + 3x\right]_1^2\]

At \(x = 2\): \(8 - 8 + 6 = 6\). At \(x = 1\): \(\tfrac{1}{2} - 2 + 3 = \tfrac{3}{2}\).

\[= 6 - \tfrac{3}{2} = \tfrac{9}{2} = 4.5\]

(b) \(P\) is the inverse of \(P^{-1}\). With \(P^{-1} = \begin{pmatrix} -1 & 1 \\ 4 & -3 \end{pmatrix}\), \(\det = (-1)(-3) - (1)(4) = -1\).

\[P = \frac{1}{-1}\begin{pmatrix} -3 & -1 \\ -4 & -1 \end{pmatrix} = \begin{pmatrix} 3 & 1 \\ 4 & 1 \end{pmatrix}\]

Check: \(P\,P^{-1} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\).

(a) Evaluate: \(\int \limits_1^2 (2x^3 - 4x + 3) dx\) (b) Given that P\(^{-1}\) = \(\begin{pmatrix} -1 & 1 \\ 4 & -3 \end{pmatrix}\), find the matrix ...