Chapter 4: Q. 389 (page 483)

In the following exercise, evaluate each determinant by expanding by minors along the first row.
$|\begin{array}{r} - 2 & - 3 & - 4 \\ 5 & - 6 & 7 \\ - 1 & 2 & 0 \end{array}|$

Short Answer

Expert verified

The value of the given determinant is $33$ .

Step by step solution

Step 1. Given information

The given determinant is :

$|\begin{array}{r} - 2 & - 3 & - 4 \\ 5 & - 6 & 7 \\ - 1 & 2 & 0 \end{array}|$

Step 2. Expanding the determinant along the first row by minors

The expansion of general determinant along the first row is given as: $|\begin{array}{l} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array}| = a_{1} |\begin{array}{l} b_{2} & c_{2} \\ b_{3} & c_{3} \end{array}| - b_{1} |\begin{array}{l} a_{2} & c_{2} \\ a_{3} & c_{3} \end{array}| + c_{1} |\begin{array}{l} a_{2} & b_{2} \\ a_{3} & b_{3} \end{array}|$

Similarly, for the given determinant, we have:

localid="1644588932923" $|\begin{matrix} - 2 & - 3 & - 4 \\ 5 & - 6 & 7 \\ - 1 & 2 & 0 \end{matrix}| = - 2 |\begin{matrix} - 6 & 7 \\ 2 & 0 \end{matrix}| + 3 |\begin{matrix} 5 & 7 \\ - 1 & 0 \end{matrix}| - 4 |\begin{matrix} 5 & - 6 \\ - 1 & 2 \end{matrix}|$

Step 3. evaluating the determinant

Evaluating the determinant, we get:

$= - 2 |\begin{matrix} - 6 & 7 \\ 2 & 0 \end{matrix}| + 3 |\begin{matrix} 5 & 7 \\ - 1 & 0 \end{matrix}| - 4 |\begin{matrix} 5 & - 6 \\ - 1 & 2 \end{matrix}| = - 2 [- 6 \times 0 - 7 \times 2] + 3 [5 \times 0 - (- 1 \times 7)] - 4 [5 \times 2 - (- 6 \times - 1)] = - 2 [0 - 14] + 3 [7] - 4 [10 - 6] = 28 + 21 - 40 = 33$

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In the following exercise, evaluate each determinant by expanding by minors along the first row.
$|\begin{array}{r} - 2 & - 3 & - 4 \\ 5 & - 6 & 7 \\ - 1 & 2 & 0 \end{array}|$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Expanding the determinant along the first row by minors

Step 3. evaluating the determinant

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In the following exercise, evaluate each determinant by expanding by minors along the first row.-2-3-45-67-120

Short Answer

Step by step solution

Step 1. Given information

Step 2. Expanding the determinant along the first row by minors

Step 3. evaluating the determinant

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Geometry

Pure Maths

Statistics

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Probability and Statistics

Study anywhere. Anytime. Across all devices.

In the following exercise, evaluate each determinant by expanding by minors along the first row.
$|\begin{array}{r} - 2 & - 3 & - 4 \\ 5 & - 6 & 7 \\ - 1 & 2 & 0 \end{array}|$