Question

evaluate the following determinants. 15. \\(\

$$\begin{vmatrix}1 & 2 \\\\ 2 & 5\\end{vmatrix}$$

\\) 16. \\(\

$$\begin{vmatrix}2 & 3 \\\\ 4 & 5\\end{vmatrix}$$

\\) 17. \\(\

$$\begin{vmatrix}3 & 2 \\\\ 6 & 4\\end{vmatrix}$$

\\) 18. \\(\

$$\begin{vmatrix}-2 & 2 \\\\ 3 & 5\\end{vmatrix}$$

\\) 19. \\(\

$$\begin{vmatrix}3 & 5 \\\\ -2 & 4\\end{vmatrix}$$

\\) 20. \\(\

$$\begin{vmatrix}7 & 5 \\\\ -3 & -2\\end{vmatrix}$$

\\)

Explanation:

Step1: Recall the formula for 2x2 determinant

For a matrix

$$\begin{vmatrix}a&b\\c&d\end{vmatrix}$$

, the determinant is $ad - bc$.

Step15: Evaluate

$$\begin{vmatrix}1&2\\2&5\end{vmatrix}$$

Using the formula: $1\times5 - 2\times2 = 5 - 4 = 1$

Step16: Evaluate

$$\begin{vmatrix}2&3\\4&5\end{vmatrix}$$

Using the formula: $2\times5 - 3\times4 = 10 - 12 = -2$

Step17: Evaluate

$$\begin{vmatrix}3&2\\6&4\end{vmatrix}$$

Using the formula: $3\times4 - 2\times6 = 12 - 12 = 0$

Step18: Evaluate

$$\begin{vmatrix}-2&2\\3&5\end{vmatrix}$$

Using the formula: $(-2)\times5 - 2\times3 = -10 - 6 = -16$

Step19: Evaluate

$$\begin{vmatrix}3&5\\-2&4\end{vmatrix}$$

Using the formula: $3\times4 - 5\times(-2) = 12 + 10 = 22$

Step20: Evaluate

$$\begin{vmatrix}7&5\\-3&-2\end{vmatrix}$$

Using the formula: $7\times(-2) - 5\times(-3) = -14 + 15 = 1$

Answer:

1. $\boldsymbol{1}$
2. $\boldsymbol{-2}$
3. $\boldsymbol{0}$
4. $\boldsymbol{-16}$
5. $\boldsymbol{22}$
6. $\boldsymbol{1}$