Find the determinant of the matrix
$\left{\begin{array}{lll}1 & -2 & 3\end{array}\right.$
A) $\begin{array}{ccc}
-2 & 1 & 2 \\
-1 & 1 & -1
\end{array}$
Evaluate the integrals
$\int_1^4\left(\frac{1}{x-1}+\frac{1}{x^2}+\frac{1}{x^3}\right) d x$
Answer (2)
The determinant of the given matrix is -6, calculated using the determinant formula for a 3x3 matrix. The integral diverges because of the presence of a singularity at the lower limit. Hence, the final results are: determinant = -6 and the integral diverges.
;
Calculate the determinant of the matrix using cofactor expansion, obtaining det ( A ) = 2 .
Recognize that the integral ∫ 1 4 x − 1 1 d x is improper due to the singularity at x = 1 .
Determine that the definite integral ∫ 1 4 ( x − 1 1 + x 2 1 + x 3 1 ) d x diverges because of the improper integral.
State the final answers: The determinant is 2 , and the integral d i v er g es .
Explanation
Problem Overview We are given two problems: finding the determinant of a 3x3 matrix and evaluating a definite integral. Let's solve them one by one.
Determinant Calculation The matrix is given as:
A = 1 − 2 − 1 − 2 1 1 3 2 − 1
To find the determinant, we can use the cofactor expansion along the first row:
det ( A ) = 1 ⋅ 1 1 2 − 1 − ( − 2 ) ⋅ − 2 − 1 2 − 1 + 3 ⋅ − 2 − 1 1 1
Calculating 2x2 Determinants Now, let's calculate the determinants of the 2x2 matrices:
1 1 2 − 1 = ( 1 ⋅ − 1 ) − ( 2 ⋅ 1 ) = − 1 − 2 = − 3
− 2 − 1 2 − 1 = ( − 2 ⋅ − 1 ) − ( 2 ⋅ − 1 ) = 2 + 2 = 4
− 2 − 1 1 1 = ( − 2 ⋅ 1 ) − ( 1 ⋅ − 1 ) = − 2 + 1 = − 1
Final Determinant Value Substitute these values back into the determinant expression:
det ( A ) = 1 ⋅ ( − 3 ) − ( − 2 ) ⋅ ( 4 ) + 3 ⋅ ( − 1 ) = − 3 + 8 − 3 = 2
Integral Setup The definite integral is given as:
∫ 1 4 ( x − 1 1 + x 2 1 + x 3 1 ) d x
We can split this integral into three separate integrals:
∫ 1 4 x − 1 1 d x + ∫ 1 4 x 2 1 d x + ∫ 1 4 x 3 1 d x
Evaluating the First Integral Let's evaluate each integral separately. Note that the first integral is improper because the integrand x − 1 1 has a singularity at x = 1 . Therefore, we need to consider the limit as x approaches 1 from the right:
lim a → 1 + ∫ a 4 x − 1 1 d x = lim a → 1 + [ ln ∣ x − 1∣ ] a 4 = lim a → 1 + ( ln ( 3 ) − ln ( a − 1 ))
Since lim a → 1 + ln ( a − 1 ) = − ∞ , the first integral diverges.
Evaluating the Second and Third Integrals Since the first integral diverges, the entire definite integral diverges. However, if we were to ignore the improper nature of the integral and proceed with the calculation, we would have:
∫ 1 4 x 2 1 d x = [ − x 1 ] 1 4 = − 4 1 − ( − 1 ) = 1 − 4 1 = 4 3
∫ 1 4 x 3 1 d x = [ − 2 x 2 1 ] 1 4 = − 2 ( 16 ) 1 − ( − 2 ( 1 ) 1 ) = − 32 1 + 2 1 = 32 − 1 + 16 = 32 15
Combining the Integrals If we were to proceed without considering the limit, we would have:
∫ 1 4 ( x − 1 1 + x 2 1 + x 3 1 ) d x = [ ln ∣ x − 1∣ − x 1 − 2 x 2 1 ] 1 4 = ( ln ( 3 ) − 4 1 − 32 1 ) − lim a → 1 + ( ln ∣ a − 1∣ − a 1 − 2 a 2 1 )
Since lim a → 1 + ln ( a − 1 ) = − ∞ , the integral diverges.
Final Answer The determinant of the matrix is 2. The definite integral ∫ 1 4 ( x − 1 1 + x 2 1 + x 3 1 ) d x diverges because the integral ∫ 1 4 x − 1 1 d x is improper and diverges.
Examples
Understanding determinants is crucial in various fields like physics and engineering. For example, in structural analysis, the determinant of a stiffness matrix determines the stability of a structure. If the determinant is zero, the structure is unstable and can collapse. Similarly, evaluating definite integrals is essential in calculating areas, volumes, and probabilities. In physics, it can be used to find the work done by a force or the total charge in a given region.
