Chapter 4: Q31. (page 186)

Evaluate each determinant using expansion by minors.
31. $|\begin{matrix} 1 & 5 & - 4 \\ - 7 & 3 & 2 \\ 6 & 3 & - 1 \end{matrix}|$

Short Answer

Expert verified

The value of the determinant $|\begin{matrix} 1 & 5 & - 4 \\ - 7 & 3 & 2 \\ 6 & 3 & - 1 \end{matrix}| = 172$ .

Step by step solution

- Decide row to use for expansion

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

$|\begin{matrix} 1 & 5 & - 4 \\ - 7 & 3 & 2 \\ 6 & 3 & - 1 \end{matrix}| = (1) |\begin{matrix} 3 & 2 \\ 3 & - 1 \end{matrix}| - 5 |\begin{matrix} - 7 & 2 \\ 6 & - 1 \end{matrix}| + (- 4) |\begin{matrix} - 7 & 3 \\ 6 & 3 \end{matrix}|$

- 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.

$\begin{matrix} |\begin{matrix} 1 & 5 & - 4 \\ - 7 & 3 & 2 \\ 6 & 3 & - 1 \end{matrix}| = (1) |\begin{matrix} 3 & 2 \\ 3 & - 1 \end{matrix}| - 5 |\begin{matrix} - 7 & 2 \\ 6 & - 1 \end{matrix}| + (- 4) |\begin{matrix} - 7 & 3 \\ 6 & 3 \end{matrix}| \\ = (- 3 - 6) - 5 (7 - 12) - 4 (- 21 - 18) \end{matrix}$

- Multiply and simplify

Further simplify the values.

$\begin{matrix} |\begin{matrix} 1 & 5 & - 4 \\ - 7 & 3 & 2 \\ 6 & 3 & - 1 \end{matrix}| = (- 3 - 6) - 5 (7 - 12) - 4 (- 21 - 18) \\ = - 9 - 5 (- 5) - 4 (- 39) \\ = - 9 + 25 + 156 \\ = 172 \end{matrix}$

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Evaluate each determinant using expansion by minors.
31. $|\begin{matrix} 1 & 5 & - 4 \\ - 7 & 3 & 2 \\ 6 & 3 & - 1 \end{matrix}|$

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.31.15−4−73263−1

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

Logic and Functions

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Geometry

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Statistics

Study anywhere. Anytime. Across all devices.

Evaluate each determinant using expansion by minors.
31. $|\begin{matrix} 1 & 5 & - 4 \\ - 7 & 3 & 2 \\ 6 & 3 & - 1 \end{matrix}|$