Answer to Question #320572 in Linear Algebra for sung jin woo

Question

2x + y = 7

x - 2y = 1

A. Write the equation in matrix form.

B. Determine the inverse of the matrix

C. Hence solve the equations.

D. x and y are matrices

$X=\begin{bmatrix} 1 & 5 \\ 3 & 7 \end{bmatrix} Y=\begin{bmatrix} 3 & 4 \\ 2 & 1 \end{bmatrix}$

Evaluate X² + Y

Answer 1

$A:\\\left[ \begin{matrix} 2& 1\\ 1& -2\\\end{matrix} \right] \left[ \begin{array}{c} x\\ y\\\end{array} \right] =\left[ \begin{array}{c} 7\\ 1\\\end{array} \right] \\B:\\\left[ \begin{matrix} 2& 1\\ 1& -2\\\end{matrix} \right] =\frac{1}{2\cdot \left( -2 \right) -1\cdot 1}\left[ \begin{matrix} -2& -1\\ -1& 2\\\end{matrix} \right] =\left[ \begin{matrix} 0.4& 0.2\\ 0.2& -0.4\\\end{matrix} \right] \\C:\\\left[ \begin{array}{c} x\\ y\\\end{array} \right] =\left[ \begin{matrix} 0.4& 0.2\\ 0.2& -0.4\\\end{matrix} \right] \left[ \begin{array}{c} 7\\ 1\\\end{array} \right] =\left[ \begin{array}{c} 3\\ 1\\\end{array} \right] \\D:\\X^2+Y=\left[ \begin{matrix} 1& 5\\ 3& 7\\\end{matrix} \right] \left[ \begin{matrix} 1& 5\\ 3& 7\\\end{matrix} \right] +\left[ \begin{matrix} 3& 4\\ 2& 1\\\end{matrix} \right] =\left[ \begin{matrix} 16& 40\\ 24& 64\\\end{matrix} \right] +\left[ \begin{matrix} 3& 4\\ 2& 1\\\end{matrix} \right] =\left[ \begin{matrix} 19& 44\\ 26& 65\\\end{matrix} \right]$