Problem 28 - Intermediate Algebra

Question

Evaluate each determinant. $$ \left|\begin{array}{ccc} 6 & 2 & 3 \\ 1 & 5 & 4 \\ 2 & 3 & 5 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is 63.

Step 1: Understanding the Determinant Formula

To find the determinant of a 3x3 matrix \[\begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix}\]use the formula:\[ a(ei − fh) − b(di − fg) + c(dh − eg) \]

Step 2: Identifying Elements of the Matrix

Label the elements of the matrix:\[\begin{vmatrix} 6 & 2 & 3 \ 1 & 5 & 4 \ 2 & 3 & 5 \end{vmatrix}\]- \(a = 6\), \(b = 2\), \(c = 3\), \(d = 1\), \(e = 5\), \(f = 4\), \(g = 2\), \(h = 3\), \(i = 5\)

Step 3: Calculating the First Term

First, calculate:\[a(ei − fh) = 6(5 \times 5 − 4 \times 3)\]Calculate inside the parenthesis first:\[5 \times 5 = 25\] and \[4 \times 3 = 12\], so \(25 - 12 = 13\).Now multiply by 6:\[6 \times 13 = 78\].

Step 4: Calculating the Second Term

Next, calculate:\[b(di − fg) = 2(1 \times 5 - 4 \times 2)\]Calculate inside the parenthesis first:\[1 \times 5 = 5\] and \[4 \times 2 = 8\], so \(5 - 8 = -3\).Now multiply by 2:\[2 \times -3 = -6\].

Step 5: Calculating the Third Term

Finally, calculate:\[c(dh − eg) = 3(1 \times 3 - 5 \times 2)\]Calculate inside the parenthesis first:\[1 \times 3 = 3\] and \[5 \times 2 = 10\], so \(3 - 10 = -7\).Now multiply by 3:\[3 \times -7 = -21\].

Step 6: Summing Up the Terms

Sum the calculated terms together:\[78 - (-6) + (-21) = 78 + 6 - 21\]Calculate step by step:\[78 + 6 = 84\]\[84 - 21 = 63\].

Key Concepts

3x3 Matrix Determinant

Understanding the determinant of a 3x3 matrix is essential in linear algebra. A determinant offers insights into properties of the matrix like singularity (whether it has an inverse), and can even provide the volume of a parallelepiped defined by the matrix's rows or columns.
For a 3x3 matrix, the formula to calculate the determinant is a bit more involved than smaller matrices but still quite approachable. With a matrix:

• \[\begin{vmatrix}a & b & c \d & e & f \g & h & i\end{vmatrix}\]

you use the expression:

• \[a(ei − fh) − b(di − fg) + c(dh − eg)\]

This formula might look complex at first, but it simply involves multiplying and subtracting a series of terms. It breaks down into manageable pieces, where each has its own minor determinant calculated by ignoring one row and column, multiplying the remaining elements, and collecting these values to find the final determinant.

Matrix Operations

Matrix operations are fundamental in both pure and applied mathematics, and they serve various purposes in solving linear equations, transforming data, and even rendering graphics in computer science. Understanding these operations allows you to manipulate matrices to achieve desired results.
Here are a few vital matrix operations:

• Addition: Combine two matrices of the same size by adding their corresponding elements.
• Subtraction: As with addition, subtract the elements of one matrix from another of the same size.
• Multiplication: This operation is more complex; you multiply each row element of the first matrix with each column element of the second, summing them up to result in a new matrix.
• Transposition: Change rows into columns, and vice versa, requiring no change in the matrix contents.

Matrix operations allow you to approach linear transformations, solve systems of linear equations, and explore higher-dimensional spaces with ease. They become smoother and more intuitive with practice, especially when combined with concepts like determinants.

Linear Algebra

Linear algebra is a branch of mathematics focusing on vector spaces, linear mappings, and the systems of linear equations. It's a powerful toolset for modeling and solving real-world problems in engineering, physics, computer graphics, and beyond.
Some core aspects and applications of linear algebra include:

• Vectors and Spaces: Understanding the representation of vectors and their manipulation within vector spaces.
• Linear Transformations: How matrices can transform vectors, exploring concepts like rotation, scaling, and reflection.
• Eigenvalues and Eigenvectors: Finding these provides profound insights into matrix behavior, stability of systems, and much more.

Linear algebra is used in dimensionality reduction, optimization problems, and proving mathematical propositions. It provides a foundation for machine learning algorithms, data analysis, and even econometrics. Mastering its concepts enables insights into both theoretical and practical problems, opening the door to many advanced topics.