Determinants Quiz 1

Question 1

If $${{M}^{2}}-4M-5{{I}_{3}}=0$$ where $$M=\left[ \begin{matrix}   1 & 2 & 2  \   2 & 1 & 2  \   2 & 2 & 1  \\end{matrix} ight]$$. Find $${{M}^{-1}}$$.

Accepted Answer

$$\dfrac{1}{5}\left[ \begin{matrix}   -3 & 2 & 2  \   2 & -3 & 2  \   2 & 2 & -3  \\end{matrix} ight]$$

Answer

$$-\dfrac{1}{5}\left[ \begin{matrix}   -3 & 2 & 2  \   2 & -3 & 2  \   2 & 2 & -3  \\end{matrix} ight]$$

Answer

$$-\dfrac{1}{10}\left[ \begin{matrix}   -3 & 2 & 2  \   2 & -3 & 2  \   2 & 2 & -3  \\end{matrix} ight]$$

Answer

$$\dfrac{1}{10}\left[ \begin{matrix}   -3 & 2 & 2  \   2 & -3 & 2  \   2 & 2 & -3  \\end{matrix} ight]$$

Question 2

Find the value of $$\left| \begin{matrix}   1 & \alpha  & {{\alpha }^{2}}-\beta \gamma   \   1 & \beta  & {{\beta }^{2}}-\alpha \gamma   \   1 & \gamma  & {{\gamma }^{2}}-\alpha \beta   \\end{matrix} ight|$$.

Accepted Answer

$$0$$

Answer

$$2$$

Answer

$$1$$

Answer

$$3$$

Question 3

Use Cramer’s rule to solve the following set of equations $$p+q+r=7,	ext{ }p+2q+3r=15,	ext{ }p-q+r=3$$

Accepted Answer

$$p=2,	ext{ }q=2,	ext{ }r=3$$

Answer

$$p=-2,	ext{ }q=-2,	ext{ }r=3$$

Answer

$$p=2,	ext{ }q=2,	ext{ }r=2$$

Answer

$$p=2,	ext{ }q=3,	ext{ }r=3$$

Question 4

Find the value of  $$\left| \begin{matrix}   {{\left( b+c ight)}^{2}} & {{c}^{2}} & {{b}^{2}}  \   {{c}^{2}} & {{\left( c+a ight)}^{2}} & {{a}^{2}}  \   {{b}^{2}} & {{a}^{2}} & {{\left( a+b ight)}^{2}}  \\end{matrix} ight|$$.

Accepted Answer

$$2{{\left( ab+bc+ca \right)}^{2}}$$

Answer

$$2\left( ab+bc+ca \right)$$

Answer

$${{\left( ab+bc+ca \right)}^{2}}$$

Answer

$$4{{\left( ab+bc+ca \right)}^{2}}$$

Question 5

If  $$\left( 1+\dfrac{1}{u}+\dfrac{1}{v}+\dfrac{1}{w}+\dfrac{1}{x} ight)$$ is one of the factors of the determinant of $$\left| \begin{matrix}   1+u & 1 & 1 & 1  \   1 & 1+v & 1 & 1  \   1 & 1 & 1+w & 1  \   1 & 1 & 1 & 1+x  \\end{matrix} ight|$$ then find the other factor.

Accepted Answer

$$uvwx$$

Answer

$$1$$

Answer

$$uwx$$

Answer

$$uvw$$