Evren Z.

The Dr. of Mathematics

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Linear Algebra

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Question:

Prove that the following matrix $$A$$ is invertible. $$\begin{bmatrix}10 & 3 \\ 4 & 5\end{bmatrix}$$.

Evren Z.

Answer:

The matrix $$A$$ is invertible $$\Leftrightarrow $$ $$det(A)\neq 0 $$. So, $$det(A)=50-12=38\neq 0\Leftrightarrow A$$ is invertible.

Geometry

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Question:

Write the equation $$x^{2}+y^{2}-3x+4y+4=0$$ in standard form.

Evren Z.

Answer:

Completing the square for the $$x$$-terms with $$\frac{9}{4}$$ and for the $$y$$-terms with 4, $$ \left( x^{2}-3x+\frac{9}{4} \right)+\left ( y^{2}+4y+4 \right)=-4+\frac{9}{4}+4$$. Factoring each group of terms gives rise to $$\left ( x-\frac{3}{2} \right )^{2}+\left ( y+2 \right )^{2}=\left ( \frac{3}{2} \right )^{2}$$.

Calculus

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Question:

Give the geometrical interpretation of the integral $$\int_{0}^{1}x^2dx$$.

Evren Z.

Answer:

Keep in mind that there is a direct link between geometry and calculus in terms of evaluating the definite integrals. The integral gives us the area bounded by the curve $$y=x^{2}$$ , the $$x-axis$$ and the line $$x=1$$.

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