Chapter 4: Q40. (page 186)

Solve $\det [\begin{matrix} 4 & x & - 2 \\ - x & - 3 & 1 \\ - 6 & 2 & 3 \end{matrix}] = - 3$ for x.

Short Answer

Expert verified

The value of $x = \frac{5}{3}, x = - 1$ .

Step by step solution

- Calculate the determinant

Find the determinant in this case by using expansion by minors.

$\begin{matrix} \det [\begin{matrix} 4 & x & - 2 \\ - x & - 3 & 1 \\ - 6 & 2 & 3 \end{matrix}] = - 3 \\ 4 |\begin{matrix} - 3 & 1 \\ 2 & 3 \end{matrix}| - x |\begin{matrix} - x & 1 \\ - 6 & 3 \end{matrix}| + (- 2) |\begin{matrix} - x & - 3 \\ - 6 & 2 \end{matrix}| = - 3 \end{matrix}$

- Evaluate 2×2 determinants

The determinant of second order matrix is found by calculating the difference of the product of the two diagonals, that is., $|\begin{matrix} a & b \\ c & d \end{matrix}| = a d - b c$ .Apply this definition to find the $2 \times 2$ determinant.

$\begin{matrix} 4 |\begin{matrix} - 3 & 1 \\ 2 & 3 \end{matrix}| - x |\begin{matrix} - x & 1 \\ - 6 & 3 \end{matrix}| + (- 2) |\begin{matrix} - x & - 3 \\ - 6 & 2 \end{matrix}| = - 3 \\ 4 (- 3 (3) - 2 (1)) - x (- x (3) - (- 6)) - 2 (- x (2) - (- 3) (- 6)) = - 3 \\ 4 (- 9 - 2) - x (- 3 x + 6) - 2 (- 2 x - 18) = - 3 \end{matrix}$

- Solve for x

Further simplify the values.

$\begin{matrix} 4 (- 9 - 2) - x (- 3 x + 6) - 2 (- 2 x - 18) = - 3 \\ 4 (- 11) - x (- 3 x) - x (6) - 2 (- 2 x) - 2 (- 18) = - 3 \\ - 44 + 3 x^{2} - 6 x + 4 x + 36 = - 3 \\ 3 x^{2} - 2 x - 5 = 0 \\ 3 x^{2} + 3 x - 5 x - 5 = 0 \\ 3 x (x + 1) - 5 (x + 1) = 0 \\ (3 x - 5) (x + 1) = 0 \\ x = \frac{5}{3}, x = - 1 \end{matrix}$

Therefore, possible values for x are $x = \frac{5}{3}, x = - 1$ .

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Solve $\det [\begin{matrix} 4 & x & - 2 \\ - x & - 3 & 1 \\ - 6 & 2 & 3 \end{matrix}] = - 3$ for x.

Short Answer

Step by step solution

- Calculate the determinant

- Evaluate 2×2 determinants

- Solve for x

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Solvedet4x−2−x−31−623=−3for x.

Short Answer

Step by step solution

- Calculate the determinant

- Evaluate 2×2 determinants

- Solve for x

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Study anywhere. Anytime. Across all devices.

Solve $\det [\begin{matrix} 4 & x & - 2 \\ - x & - 3 & 1 \\ - 6 & 2 & 3 \end{matrix}] = - 3$ for x.