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Chapter 9: Problem 70

Use the determinant theorems to find the value of each determinant. $$\left|\begin{array}{rrr} 9 & 1 & 7 \\ 12 & 5 & 2 \\ 11 & 4 & 3 \end{array}\right|$$

Short Answer

Expert verified
The determinant is 0.

Step by step solution

01

- Identify the matrix

Recognize the given matrix: $$\begin{array}{ccc} 9 & 1 & 7 \ 12 & 5 & 2 \ 11 & 4 & 3 \end{array}$$
02

- Apply the determinant formula for a 3x3 matrix

Use the formula for the determinant of a 3x3 matrix: $$\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$$ Where for the matrix $$\begin{array}{ccc} a & b & c \ d & e & f \ g & h & i\end{array}$$, the corresponding elements in our matrix are: a = 9, b = 1, c = 7 d = 12, e = 5, f = 2 g = 11, h = 4, i = 3
03

- Substitute the values into the formula

Substitute the values into the determinant formula: $$\text{det}(A) = 9(5 \times 3 - 2 \times 4) - 1(12 \times 3 - 2 \times 11) + 7(12 \times 4 - 5 \times 11)$$ Simplify the expressions inside the parentheses: $$5 \times 3 = 15$$ $$2 \times 4 = 8$$ $$12 \times 3 = 36$$ $$2 \times 11 = 22$$ $$12 \times 4 = 48$$ $$5 \times 11 = 55$$
04

- Perform the multiplications and subtractions

Evaluate the expressions inside the parentheses: $$15 - 8 = 7$$ $$36 - 22 = 14$$ $$48 - 55 = -7$$ Substitute back into the determinant formula: $$\text{det}(A) = 9(7) - 1(14) + 7(-7)$$
05

- Final calculation

Perform the final calculations: $$9 \times 7 = 63$$ $$1 \times 14 = 14$$ $$7 \times -7 = -49$$ Add and subtract the results: $$\text{det}(A) = 63 - 14 - 49 = 0$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

3x3 Matrix
A 3x3 matrix is a rectangular array of numbers arranged in three rows and three columns. It looks like this:
\[ \begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \ \ \ \ \end{array} \]
Each number in the matrix is called an element. In our exercise, the given matrix is:
\[ \begin{array}{ccc} 9 & 1 & 7 \ 12 & 5 & 2 \ 11 & 4 & 3 \ \ \ \ \end{array} \]
The elements are indexed by their row and column positions. For example, the element in the first row and second column is 1. Understanding matrix notation is the first step in solving problems that involve matrices.
Determinant Formula
The determinant of a 3x3 matrix is a special number that can be calculated using a specific formula. Given a matrix:
\[ \begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \ \ \ \ \end{array} \]
The formula for the determinant is:
\[ det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
In our matrix, the corresponding elements are:
\[ a = 9, b = 1, c = 7 \] \[ d = 12, e = 5, f = 2 \] \[ g = 11, h = 4, i = 3 \]
By substituting these values into the determinant formula, we can calculate the determinant step by step. This formula helps simplify the complex problem of finding the determinant.
Matrix Algebra
Matrix algebra involves a set of operations that can be performed on matrices, such as addition, subtraction, and multiplication. One important concept in matrix algebra is the determinant, which we've calculated using the formula.
Understanding matrix algebra is crucial for solving systems of linear equations, performing linear transformations, and more.
When using the determinant formula, you are applying matrix algebra techniques. This method breaks down complex interactions between matrix elements into manageable parts. This step-by-step process ensures accuracy and provides insight into the deeper structure of the matrix.
Elementary Row Operations
Elementary row operations are techniques used to manipulate matrices to solve systems of equations, find determinants, and more. These operations include:
  • Swapping two rows
  • Multiplying a row by a non-zero scalar
  • Adding or subtracting a multiple of one row to another row

In our exercise, while we did not explicitly use these operations to calculate the determinant, they are often used in other matrix-related procedures. Understanding these operations can aid in simplifying matrices and making determinant calculation more intuitive.
Elementary row operations are critical for techniques like Gaussian elimination, which simplifies matrices to find solutions to linear systems or to find determinants by transforming the matrix into an upper triangular form.

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Most popular questions from this chapter

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