Question

Solve for \(x\) \(\left|\begin{array}{rrr}3 & 2 & 4 \\ 1 & x & 5 \\ 0 & 1 & -2\end{array}\right|=0\)

Short answer

x = -\frac{11}{6}

Step-by-step solution

- Understand the Determinant of a Matrix

The problem requires solving for the variable \(x\) by setting the determinant of a 3x3 matrix to zero. The matrix given is: i\[\begin{bmatrix} 3 & 2 & 4 \ 1 & x & 5 \ 0 & 1 & -2 \end{bmatrix} = 0 \]

- Formula for Determinant

The determinant of a 3x3 matrix \(\begin{{bmatrix}} a & b & c \ d & e & f \ g & h & i \end{{bmatrix}} \) is calculated as:\[ \text{Determinant} = a(ei - fh) - b(di - fg) + c(dh - eg) \]

- Substitute the Values

Substitute the values from the matrix into the determinant formula: \[ 3(x(-2) - 5(1)) - 2(1(-2) - 5(0)) + 4(1(1) - x(0)) \]

- Simplify the Expression

Simplify each term in the expression: \[ 3(-2x - 5) - 2(-2) + 4(1) \ = -6x - 15 + 4 + 4 \ = -6x - 11 \]

- Set the Equation to Zero

Set the simplified expression equal to zero: \[ -6x - 11 = 0 \]

- Solve for \( x \)

Solve the equation for \( x \): \[ -6x = 11 \approx x = -\frac{11}{6} \]

Key concepts

3x3 Matrix

A 3x3 matrix is a grid of numbers arranged in three rows and three columns. Each position in the grid holds a number, making a total of nine elements. For example, in our problem, the matrix is: \[\begin{bmatrix} 3 & 2 & 4 \ 1 & x & 5 \ 0 & 1 & -2 \end{bmatrix}\]. Each element's position is crucial for solving matrix equations, especially when we need to compute the determinant.

Matrix Determinant Formula

To find the determinant of a 3x3 matrix \(\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \ \end{bmatrix}\), we use the formula:
\[ \text{Determinant} = a(ei - fh) - b(di - fg) + c(dh - eg) \]
This equation combines the products of different elements in the matrix. Each term in the formula corresponds to a specific combination of elements, where you multiply the numbers in diagonal patterns (both forward and backward), then sum and subtract these products as specified. This method ensures that we account for all parts of the matrix in a structured way.

Solving Determinants

To solve for the variable \(x\) in our matrix determinant, follow these steps:
First, identify the elements in the matrix and plug them into the determinant formula:
\[ 3(x(-2) - 5(1)) - 2(1(-2) - 5(0)) + 4(1(1) - x(0)) \]
Next, simplify each part of the expression step-by-step to get:
\[ 3(-2x - 5) - 2(-2) + 4(1) = -6x - 15 + 4 + 4 = -6x - 11 \]
Finally, set the simplified expression to zero and solve for \(x\):
\[ -6x - 11 = 0 \]
This results in: \[ -6x = 11 \]
Therefore, \( x = -\frac{11}{6} \).
Make sure to double-check each multiplication and addition step to avoid any mistakes in calculation.