Chapter 4: Q32. (page 186)

Evaluate each determinant using expansion by minors.
$|\begin{matrix} 3 & 7 & 6 \\ - 1 & 6 & 2 \\ 8 & - 3 & - 5 \end{matrix}|$

Short Answer

Expert verified

The value of the determinant $|\begin{matrix} 3 & 7 & 6 \\ - 1 & 6 & 2 \\ 8 & - 3 & - 5 \end{matrix}| = - 265$ .

Step by step solution

- Decide row to use for expansion

In this case, use first row for the expansion by minors.

$|\begin{matrix} 3 & 7 & 6 \\ - 1 & 6 & 2 \\ 8 & - 3 & - 5 \end{matrix}| = 3 |\begin{matrix} 6 & 2 \\ - 3 & - 5 \end{matrix}| - 7 |\begin{matrix} - 1 & 2 \\ 8 & - 5 \end{matrix}| + 6 |\begin{matrix} - 1 & 6 \\ 8 & - 3 \end{matrix}|$

- 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.

$\begin{matrix} |\begin{matrix} 3 & 7 & 6 \\ - 1 & 6 & 2 \\ 8 & - 3 & - 5 \end{matrix}| = 3 |\begin{matrix} 6 & 2 \\ - 3 & - 5 \end{matrix}| - 7 |\begin{matrix} - 1 & 2 \\ 8 & - 5 \end{matrix}| + 6 |\begin{matrix} - 1 & 6 \\ 8 & - 3 \end{matrix}| \\ = 3 (- 30 - (- 6)) - 7 (5 - 16) + 6 (3 - 48) \end{matrix}$

- Multiply and simplify

Further simplify the values.

$\begin{matrix} |\begin{matrix} 3 & 7 & 6 \\ - 1 & 6 & 2 \\ 8 & - 3 & - 5 \end{matrix}| = 3 (- 30 - (- 6)) - 7 (5 - 16) + 6 (3 - 48) \\ = 3 (- 24) - 7 (- 11) + 6 (- 45) \\ = - 72 + 77 - 270 \\ = - 265 \end{matrix}$

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Evaluate each determinant using expansion by minors.
$|\begin{matrix} 3 & 7 & 6 \\ - 1 & 6 & 2 \\ 8 & - 3 & - 5 \end{matrix}|$

Short Answer

Step by step solution

- Decide row to use for expansion

- Evaluate 2×2 determinants

- Multiply and simplify

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Evaluate each determinant using expansion by minors.376−1628−3−5

Short Answer

Step by step solution

- Decide row to use for expansion

- Evaluate 2×2 determinants

- Multiply and simplify

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Logic and Functions

Calculus

Theoretical and Mathematical Physics

Probability and Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Evaluate each determinant using expansion by minors.
$|\begin{matrix} 3 & 7 & 6 \\ - 1 & 6 & 2 \\ 8 & - 3 & - 5 \end{matrix}|$