Given System of Equations:

We are given the matrix equation:

[ \left[\begin{array}{cc} 7 & 27 \ 9 & -11 \end{array}\right] \left[\begin{array}{c} x \ y \end{array}\right] = \left[\begin{array}{c} 5 \ 2 \end{array}\right] ]

Step 1: Row Reduction

To solve for $x$ and $y$, we will perform row reduction (Gaussian elimination). The goal is to transform the system into an upper triangular form so that back substitution can be used to solve for $x$ and $y$.

The system is:

$7x + 27y = 5$ $9x - 11y = 2$

To eliminate $x$ from the second equation, we can perform the following row operation:

$R_2 \rightarrow R_2 - \left(\frac{9}{7}\right)R_1$

Step-by-Step Row Operation:

1. The first equation remains the same: $7x + 27y = 5$

2. Subtract $\frac{9}{7}$ times the first row from the second row: $9x - 11y - \left(\frac{9}{7}\right)(7x + 27y) = 2 - \left(\frac{9}{7}\right)(5)$

Simplify:

$9x - 11y - \left(9x + \frac{243}{7}y\right) = 2 - \frac{45}{7}$

This simplifies to:

$9x - 11y - 9x - \frac{243}{7}y = \frac{14}{7} - \frac{45}{7}$

$-11y - \frac{243}{7}y = \frac{-31}{7}$

Multiply through by 7 to simplify:

$-77y - 243y = -31$

$-320y = -31$

Solve for $y$:

$y = \frac{-31}{-320} = \frac{31}{320}$

Step 2: Back Substitution

Now that we have $y = \frac{31}{320}$, we substitute this value into the first equation to solve for $x$:

$7x + 27\left(\frac{31}{320}\right) = 5$

Simplify:

$7x + \frac{837}{320} = 5$

Multiply both sides by 320 to eliminate the fraction:

$2240x + 837 = 1600$

Solve for $x$:

$2240x = 1600 - 837 = 763$

$x = \frac{763}{2240}$

Thus, the solution is: $x = \frac{763}{2240}, \quad y = \frac{31}{320}$

Step 3: Calculate the Determinant

The determinant of the original matrix is:

$\text{det} = (7)(-11) - (9)(27)$

$\text{det} = -77 - 243 = -320$

Final Answer:

$x = \frac{763}{2240}, \quad y = \frac{31}{320}, \quad \text{determinant} = -320$

Would you like further details or have any questions?

Here are some related questions:

1. How does Gaussian elimination differ from row reduction?
2. Can the determinant of a matrix ever be zero?
3. How are type 3 row operations used in matrix reduction?
4. What is the significance of the determinant in a system of equations?
5. How do row operations affect the determinant of a matrix?

Tip: When the determinant of a matrix is zero, the system of equations has either no solution or infinitely many solutions.