by Guest » Sun Nov 29, 2015 6:37 pm
i) [tex]\pmatrix{2 & 3 \\ 1 & -2}\pmatrix{x_1\\x_2}=\pmatrix{1\\4}[/tex]
ii) [tex]\begin{vmatrix}2 & 3 \\1 & -2\end{vmatrix}= (2\times -2) - (3\times 1) = -7[/tex]
iii) Cramer's rule says to find the value of [tex]x_i[/tex] replace column [tex]i[/tex] in [tex]A[/tex] with [tex]\mathbf{b}[/tex] and divide the determinant of this new matrix by [tex]\det(A)[/tex].
[tex]x_1 = \begin{vmatrix}\mathbf{1} & 3 \\ \mathbf{4} & -2\end{vmatrix} / \det(A) = ((1\times -2) - (3\times 4))/-7 = 2[/tex]
[tex]x_2 = \begin{vmatrix} 2 &\mathbf{1} \\ 1 & \mathbf{4} \end{vmatrix} / \det(A) = ((2\times 4) - (1\times 1))/-7 = -1[/tex]
(Once you have the solution you should always double check you didn't make a mistake by substituting it back into the equations, which is easy to do.)
Hope this helped,
R. Baber.