Problem 43 - Precalculus: Functions and Graphs

Question

Find the determinant of the matrix. $$\left[\begin{array}{rrr} 29 & -17 & 90 \\ -34 & 91 & -34 \\ 48 & 7 & 10 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of the matrix is -407924.

Step 1: Identify the Matrix

The given matrix is a 3x3 matrix:\[\left[ \begin{array}{rrr} 29 & -17 & 90 \ -34 & 91 & -34 \ 48 & 7 & 10 \end{array} \right]\]

Step 2: Apply the Formula for Determinant of a 3x3 Matrix

The determinant \( \det(A) \) of a 3x3 matrix \(\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\)is calculated as:\[\det(A) = a(ei-fh) - b(di-fg) + c(dh-eg)\]

Step 3: Substitute the Values into the Formula

Using the elements of the given matrix:\( a = 29, b = -17, c = 90, d = -34, e = 91, f = -34, g = 48, h = 7, i = 10 \).Substitute these values into the formula:\[\det(A) = 29(91 \times 10 - (-34) \times 7) - (-17)(-34 \times 10 - (-34) \times 48) + 90(-34 \times 7 - 91 \times 48)\]

Step 4: Calculate Individual Terms

Calculate each part:1. \( ei - fh = 91 \times 10 - (-34) \times 7 = 910 + 238 = 1148 \).2. \( di - fg = -34 \times 10 - (-34) \times 48 = -340 + 1632 = 1292 \).3. \( dh - eg = -34 \times 7 - 91 \times 48 = -238 - 4368 = -4606 \).

Step 5: Compute the Determinant

Now, substitute back to compute the determinant:\[\det(A) = 29 \times 1148 + 17 \times 1292 + 90 \times (-4606)\]Evaluating further:\[= 33292 + 21964 - 414540 = 7616 - 414540 = -407924\]

Step 6: Conclude the Calculation

The determinant of the matrix is calculated as \(-407924\).

Key Concepts

3x3 Matrix

In mathematics, a 3x3 matrix is a rectangular array consisting of 3 rows and 3 columns of elements. In the case of our exercise, the matrix is:\[\begin{bmatrix}29 & -17 & 90 \-34 & 91 & -34 \48 & 7 & 10\end{bmatrix}\]A matrix like this is often used to represent data or to perform transformations in algebra. The size of the matrix, in this case "3x3", indicates that it has three rows and three columns. This specific structure allows for various mathematical operations to be performed, such as finding the determinant or multiplying with other matrices. Understanding the arrangement and structure of a matrix is crucial for further operations in linear algebra.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, linear mappings, and systems of linear equations. It's essential in solving real-world problems involving multiple variables and equations. At its core, linear algebra provides the language and rules for working with matrices, like our 3x3 matrix example. Some key concepts in linear algebra include:

• Vectors: Ordered lists of numbers that can represent points or directions in space.
• Vector Spaces: Collections of vectors that can be added together and scaled.
• Transformations: Functions or operations that change vectors while preserving their vector space structure.

In the exercise at hand, linear algebra helps determine the determinant of a matrix, which is a value indicating specific properties of the matrix, such as invertibility.

Matrix Operations

Matrix operations are procedures that involve manipulating matrices in various ways, such as addition, subtraction, multiplication, and finding determinants. For our specific exercise, we focused on finding the determinant of a 3x3 matrix, which is a special operation providing important information about the matrix.The determinant is a scalar value that can reveal if the matrix can be inverted or if the related system of linear equations has a unique solution.To find the determinant for a 3x3 matrix, we use the formula:\[det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]Here are the steps to find the determinant:

• Identify each element from the matrix: a, b, c (top row), d, e, f (middle row), g, h, i (bottom row).
• Substitute those elements into the formula.
• Calculate each term separately: \(ei-fh\), \(di-fg\), \(dh-eg\).
• Combine these results to get the determinant of the matrix.

Finding the determinant is just one of many operations you can perform with matrices, each serving different purposes in mathematical and real-world applications.