The problem asks to calculate the determinants of the following 2x2 matrices: a) $\begin{vmatrix} 1 & 2 \\ 2 & 5 \end{vmatrix}$ b) $\begin{vmatrix} 1 & 3 \\ 1 & 7 \end{vmatrix}$ c) $\begin{vmatrix} 5 & 7 \\ 6 & 8 \end{vmatrix}$ d) $\begin{vmatrix} 3 & 6 \\ 4 & 8 \end{vmatrix}$ e) $\begin{vmatrix} a & b \\ b & a \end{vmatrix}$ f) $\begin{vmatrix} a-b & 1 \\ 1 & a+b \end{vmatrix}$ g) $\begin{vmatrix} a-b & -2 \\ ab & a-b \end{vmatrix}$ h) $\begin{vmatrix} x & y^2 \\ y & x^2 \end{vmatrix}$

1. Problem Description

The problem asks to calculate the determinants of the following 2x2 matrices:

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

b) $\begin{vmatrix} 1 & 3 \\ 1 & 7 \end{vmatrix}$

c) $\begin{vmatrix} 5 & 7 \\ 6 & 8 \end{vmatrix}$

d) $\begin{vmatrix} 3 & 6 \\ 4 & 8 \end{vmatrix}$

e) $\begin{vmatrix} a & b \\ b & a \end{vmatrix}$

f) $\begin{vmatrix} a-b & 1 \\ 1 & a+b \end{vmatrix}$

g) $\begin{vmatrix} a-b & -2 \\ ab & a-b \end{vmatrix}$

h) $\begin{vmatrix} x & y^2 \\ y & x^2 \end{vmatrix}$

2. Solution Steps

The determinant of a 2x2 matrix $\begin{vmatrix} a & b \\ c & d \end{vmatrix}$ is calculated as $ad - bc$.

a) $\begin{vmatrix} 1 & 2 \\ 2 & 5 \end{vmatrix} = (1)(5) - (2)(2) = 5 - 4 = 1$

b) $\begin{vmatrix} 1 & 3 \\ 1 & 7 \end{vmatrix} = (1)(7) - (3)(1) = 7 - 3 = 4$

c) $\begin{vmatrix} 5 & 7 \\ 6 & 8 \end{vmatrix} = (5)(8) - (7)(6) = 40 - 42 = -2$

d) $\begin{vmatrix} 3 & 6 \\ 4 & 8 \end{vmatrix} = (3)(8) - (6)(4) = 24 - 24 = 0$

e) $\begin{vmatrix} a & b \\ b & a \end{vmatrix} = (a)(a) - (b)(b) = a^2 - b^2$

f) $\begin{vmatrix} a-b & 1 \\ 1 & a+b \end{vmatrix} = (a-b)(a+b) - (1)(1) = a^2 - b^2 - 1$

g) $\begin{vmatrix} a-b & -2 \\ ab & a-b \end{vmatrix} = (a-b)(a-b) - (-2)(ab) = (a-b)^2 + 2ab = a^2 - 2ab + b^2 + 2ab = a^2 + b^2$

h) $\begin{vmatrix} x & y^2 \\ y & x^2 \end{vmatrix} = (x)(x^2) - (y^2)(y) = x^3 - y^3$

3. Final Answer

a) 1

b) 4

c) -2

d) 0

e) $a^2 - b^2$

f) $a^2 - b^2 - 1$

g) $a^2 + b^2$

h) $x^3 - y^3$