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The determinant of a matrix is a special number that can be calculated from its elements. In the context of a 2x2 matrix, which is the simplest form beyond a 1x1 matrix, the calculation is straightforward. To find the determinant of a matrix such as \[\begin{equation}\left| \begin{array}{cc}a & b \ c & d\end{array} \right|,d\end{equation}\] we use the formula \[\begin{equation}ad - bc.d\end{equation}\] This formula is derived from a set of linear equations that the matrix represents. By substituting the matrix's elements into the formula, the determinant is the difference between the product of its diagonals. Hence, the determinant serves as an indicator of certain matrix properties, such as whether the matrix is invertible or not. A zero determinant, as found in the exercise, suggests that the matrix does not have an inverse and that the system of equations it represents may not have a unique solution.

Matrices are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. They are indispensable tools in mathematical concepts including systems of linear equations, transformations, and more. Each element within a matrix is identified by its position, indicated by two indices: the row number and the column number, respectively.

Matrices offer a compact way of representing and operating on linear transformations, and they are central in the study of linear algebra. Determinants play a crucial role in understanding these transformations, as they provide information about the 'scaling factor' of the matrix when it is considered as a linear transformation.

Elementary linear algebra is the branch of mathematics that deals with vectors, vector spaces, systems of linear equations, and matrices. It is foundational for advanced mathematical studies and applications across various scientific fields like physics, engineering, computer science, and economics.

Key Concepts in Elementary Linear Algebra:

• Vectors: Objects that have both a magnitude and a direction, represented as an array of numbers.
• Vector Spaces: A collection of vectors that can be scaled and added together following certain rules.
• Linear Equations: Equations that describe a straight line, plane, or hyperplane when plotted in n-dimensional space.
• Matrices: The representation of linear transformations and systems of linear equations.
• Determinants: A scalar attribute of a matrix that provides critical information about the matrix, such as solvability of the corresponding linear system.

All these elements are interconnected, making the subject a fascinating and rich field of study that has profound implications in both theory and application.