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A 3x3 matrix is a type of matrix that contains three rows and three columns. It is an essential tool in linear algebra, often used to describe transformations in three-dimensional space. In our exercise, the given 3x3 matrix is made up of three row vectors, with the first row consisting of the basis vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\). This is particularly common in vector algebra where orientation in space is important, such as in physics and engineering.
3x3 matrices can be used to perform operations like rotations and scaling in 3D space. They are crucial in computer graphics to manipulate the position and orientation of objects. Moreover, such matrices often help when performing calculations involving cross products and finding determinants, as is the case in this exercise. Mastering 3x3 matrices is key to understanding more complex concepts in higher-dimensional linear algebra.

Cofactor expansion, also known as Laplace's expansion, is a method used to calculate the determinant of a matrix. For a 3x3 matrix, this involves expanding the determinant along any row or column. In this exercise, we've chosen to expand along the first row, which consists of the basis vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\).
To use cofactor expansion:

• Choose a row or column (in this case, the first row).
• Compute the minor for each element of the chosen row or column. Minors are the determinants of the 2x2 matrices left after removing the row and column of the element.
• Apply the sign pattern to each minor, usually given by \((-1)^{i+j}\) where \(i\) and \(j\) are the indices of the element in the matrix.
• Sum up the product of each element and its cofactor (signed minor) to find the determinant.

This method systematically breaks down the determinant calculation into manageable parts, making it easier to solve complex problems step by step.

The calculation of a 2x2 determinant is an important step when using cofactor expansion. For any 2x2 matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), its determinant is calculated using the rule:
\[\text{Determinant} = ad - bc\]This simple formula involves multiplying across the main diagonal (\(ad\)) and then subtracting the product of the off-diagonal (\(bc\)).
In our exercise, computing 2x2 determinants helps us break down the 3x3 determinant into smaller parts. When we calculate using cofactor expansion along the row of basis vectors, we evaluate each position's minor as a 2x2 determinant. This technique ensures precision and clarity while solving complex matrix problems. By practicing this basic yet fundamental rule of determinant calculation, students can gain confidence in handling larger matrices.

Vector algebra is a branch of mathematics that deals with quantities that have both a magnitude and a direction. It is frequently used in physics and engineering to model forces, velocities, and other dynamic phenomena.
In vector algebra, one common operation is the vector cross product, which results in a vector perpendicular to two given vectors. This concept is directly related to determinant calculations of matrices involving basis vectors such as \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\). The determinant of a matrix involving these basis vectors can often represent the volume of a parallelepiped spanned by the vectors.
In our exercise, the result of the determinant calculation is a vector represented by \(21\mathbf{i} - 4\mathbf{j} - 3\mathbf{k}\). This reflects both the orientation and magnitude of the result, consistent with the rules of vector algebra. By understanding how determinant calculations tie into vector algebra, students can better appreciate the power and applicability of these mathematical concepts.