\(\left|\begin{array}{ccc}1 & \mathrm{a} & \mathrm{a}^{2}-\mathrm{bc} \\ 1 & \mathrm{~b} & \mathrm{~b}^{2}-\mathrm{ca} \\ 1 & \mathrm{c} & \mathrm{c}^{2}-\mathrm{ab}\end{array}\right|=\ldots\) (a) 0 (b) \(\left(a^{2}-b c\right)\left(b^{2}-c a\right)\left(c^{2}-a b\right)\) (c) \((a-b)(b-c)(c-a)\) (d) \(-1\)

Understanding the concept of a matrix determinant is crucial for various mathematical applications, including solving systems of linear equations and analyzing the properties of matrices. The determinant of a square matrix is a scalar value that summarizes certain characteristics of the matrix. It is computed from the elements of the matrix in a particular way and can tell us, among other things, if the matrix is invertible. A matrix that has a determinant of zero is considered singular, meaning it lacks an inverse. For example, consider the determinant of a 2x2 matrix:

\[ \begin{vmatrix} a & b \ c & d \end{vmatrix} = ad - bc \.
\] The pattern of multiplying and subtracting is more complex as the size of the matrix increases.

Expanding a determinant, also known as determinant expansion by minors and cofactors, is a method used to calculate the determinant of larger square matrices. The process involves breaking down the determinant of an \(n \times n\) matrix into smaller determinants, which are more manageable to compute. This is done by choosing a row or column (typically the one with the most zeros, to simplify calculations) and expanding along that row or column. The expansion uses the concept of minors and cofactors; the minor of an element is the determinant of the smaller matrix formed by removing the element's row and column, while a cofactor also includes a sign based on the element's position. The original determinant is the sum of the products of the elements and their corresponding cofactors. As seen in the step-by-step solution, expanding along the first row simplifies the original 3x3 determinant into three 2x2 determinants.

The algebraic properties of determinants are important for understanding how they behave under various matrix operations. Some key properties include the following:

• The determinant of a matrix is equal to the determinant of its transpose.
• The determinant of a matrix changes sign if two rows (or columns) are interchanged.
• The determinant of a matrix is zero if two rows (or columns) are identical or proportional to one another.
• When multiplying a row (or column) of a matrix by a scalar, the determinant of the matrix is multiplied by that scalar.
• The determinant of a product of matrices equals the product of their determinants.

These properties can be applied when calculating or simplifying determinants, as evidenced in complex problems where rows or columns can be manipulated to make the expansion process easier. Understanding the algebraic nature of determinants can offer shortcuts to finding solutions, preventing unnecessary lengthy calculations.