Problem 40 - Algebra and Trigonometry with Analytic Geometry

Question

\(\left|\begin{array}{rrr}i & j & k \\ 1 & -2 & 3 \\ 2 & 1 & -4\end{array}\right|\)

Step-by-Step Solution

Answer

The cross product is \(5\mathbf{i} + 10\mathbf{j} + 5\mathbf{k}\).

Step 1: Expand the determinant using cofactor expansion along the first row

\(\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & -2 & 3 \\ 2 & 1 & -4 \end{vmatrix} = \mathbf{i}[(-2)(-4)-(3)(1)] - \mathbf{j}[(1)(-4)-(3)(2)] + \mathbf{k}[(1)(1)-(-2)(2)]\)

Step 2: Compute each component

\(\mathbf{i}(8-3) - \mathbf{j}(-4-6) + \mathbf{k}(1+4) = 5\mathbf{i} + 10\mathbf{j} + 5\mathbf{k}\)

Key Concepts

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with the study of matrices and the operations that can be performed on them. Understanding matrices is crucial because they are used to represent systems of linear equations and transform space in various applications. In essence, a matrix is a rectangular array of numbers arranged in rows and columns.

Matrices can be added, subtracted, multiplied, and more. Some common operations you might perform include:

• Addition and Subtraction: Adding or subtracting matrices is done element-wise, meaning corresponding elements are added or subtracted.
• Multiplication: Matrix multiplication is not as straightforward as the previous operations. It involves a series of dot products and results in a new matrix whose dimensions may differ from the original matrices.
• Scalar Multiplication: Multiplying by a constant scalar changes each element of the matrix by that scalar.

Understanding these operations allows us to manipulate matrices to solve various mathematical problems, including those involving vectors and determinants, as seen in exercises like the one with the 3x3 matrix determinant.

Determinant Calculation

Calculating the determinant of a matrix can provide valuable insights, such as a matrix's invertibility, area scaling in transformations, and more. For a 3x3 matrix, the formula for determinant calculation involves the three rows or columns taken from the matrix.

To find the determinant of a 3x3 matrix, such as \(\begin{vmatrix} i & j & k \ 1 & -2 & 3 \ 2 & 1 & -4 \end{vmatrix}\),we apply the following steps:
- Find the "cofactor expansion" across one row or column, usually the first row for simplicity.- Multiply each element in that row by the determinant of the 2x2 matrix that remains after removing the row and column of that element.
Using the formula provided in the exercise solution, we compute this as following:\[\text{det}(A) = i ((-2)\times(-4) - 3\times1) - j(1\times(-4) - 3\times2) + k(1\times1 - (-2)\times2)\]Simplifying each term and combining them will give us the determinant value.
Understanding how to calculate a determinant is essential for solving these mathematical problems effectively.

Vector Notation

Vectors are fundamental components in both physics and mathematics. They extend real number concepts to more dimensions by considering magnitude and direction.

In the context of matrices, a vector can form part of a row or column. Typically, vectors are noted as either column vectors or row vectors, depending on the arrangement. For example, the column vectors in our 3x3 determinant example are\[a = \begin{pmatrix}i \ 1 \ 2\end{pmatrix}, b = \begin{pmatrix}j \ -2 \ 1\end{pmatrix}, c = \begin{pmatrix}k \ 3 \ -4\end{pmatrix}\].

Vectors provide a convenient way to express linear transformations and perform mathematical operations. They are often used for describing points and directions in space. In matrix algebra, the interaction of these vectors—through operations like dot products, cross products, and scalar multiples—forms the basis for more complex operations, like determinant calculations.
Learning vector notation is thus crucial for delving deeply into matrix algebra and related fields.