Chapter 4: Q27. (page 186)

Evaluate each determinant using expansion by minors.
$|\begin{matrix} 3 & 1 & 2 \\ 0 & 6 & 4 \\ 2 & 5 & 1 \end{matrix}|$

Short Answer

Expert verified

The value of the determinant $|\begin{matrix} 3 & 1 & 2 \\ 0 & 6 & 4 \\ 2 & 5 & 1 \end{matrix}| = - 58$ .

Step by step solution

- Decide row to use for expansion

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

$|\begin{matrix} 3 & 1 & 2 \\ 0 & 6 & 4 \\ 2 & 5 & 1 \end{matrix}| = 3 |\begin{matrix} 6 & 4 \\ 5 & 1 \end{matrix}| - (1) |\begin{matrix} 0 & 4 \\ 2 & 1 \end{matrix}| + 2 |\begin{matrix} 0 & 6 \\ 2 & 5 \end{matrix}|$

- 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.

$\begin{matrix} |\begin{matrix} 3 & 1 & 2 \\ 0 & 6 & 4 \\ 2 & 5 & 1 \end{matrix}| = 3 |\begin{matrix} 6 & 4 \\ 5 & 1 \end{matrix}| - (1) |\begin{matrix} 0 & 4 \\ 2 & 1 \end{matrix}| + 2 |\begin{matrix} 0 & 6 \\ 2 & 5 \end{matrix}| \\ = 3 (6 - 20) - (1) (0 - 8) + 2 (0 - 12) \end{matrix}$

- Multiply and simplify

Further simplify the values.

$\begin{matrix} |\begin{matrix} 3 & 1 & 2 \\ 0 & 6 & 4 \\ 2 & 5 & 1 \end{matrix}| = 3 (6 - 20) - (1) (0 - 8) + 2 (0 - 12) \\ = 3 (- 14) - (- 8) + 2 (- 12) \\ = - 42 + 8 - 24 \\ = - 58 \end{matrix}$

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Evaluate each determinant using expansion by minors.
$|\begin{matrix} 3 & 1 & 2 \\ 0 & 6 & 4 \\ 2 & 5 & 1 \end{matrix}|$

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.312064251

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

Geometry

Probability and Statistics

Applied Mathematics

Pure Maths

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.

Evaluate each determinant using expansion by minors.
$|\begin{matrix} 3 & 1 & 2 \\ 0 & 6 & 4 \\ 2 & 5 & 1 \end{matrix}|$