If $$ A=\left|\begin{array}{cc} 1 & 2 \\ -2 & -1 \end{array}\right| $$ and \(\phi(\mathrm{x})=(\mathrm{I}+\mathrm{x})(\mathrm{I}-\mathrm{x})^{-1}\). then \(\phi(\mathrm{A})=\ldots\) (a) \(\left|\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right|\) $$ \text { (b) }\left|\begin{array}{cc} 0 & 1 \\ -1 & -1 \end{array}\right| $$ (c) \(\left|\begin{array}{cc}1 & 1 \\ -1 & 0\end{array}\right|\) (d) \(\left|\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right|\)

The determinant of a matrix is a special scalar value that can indicate whether the matrix has an inverse, reflect the volume scaling factor of the linear transformation represented by the matrix, and have other interpretations in different contexts. To compute the determinant of a square matrix, various methods can be used, including expansion by minors or row reduction.

For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant, often represented as \( |A| \), is calculated as \( ad - bc \). In more complex matrices, the process might involve breaking down the matrix into smaller sections and calculating the determinant recursively.

Calculating the determinant is an essential step when finding the inverse of a matrix, as a matrix can only have an inverse if its determinant is non-zero.

The inverse of a matrix \( A \) is another matrix, denoted as \( A^{-1} \) and defined such that when \( A \) is multiplied by \( A^{-1} \) the result is the identity matrix. The inverse matrix is critical in solving systems of linear equations, finding matrix representations of geometric transformations, and other fields of higher mathematics.

To find the inverse of a 2x2 matrix, the procedure involves transposing the positions of the elements on the main diagonal, changing the signs of the off-diagonal elements, and then dividing each by the determinant of the original matrix. For larger matrices, the process can be more involved, sometimes requiring the use of elementary row operations or advanced algorithms.

It is important to note not every matrix has an inverse—a matrix must be square (same number of rows and columns) and have a non-zero determinant to be invertible.

The identity matrix, often represented as \( I \), is a square matrix in which all the elements on the main diagonal are 1, while all other elements are 0. It acts much like the number 1 in scalar multiplication, meaning that any matrix multiplied by the identity matrix will result in that same original matrix.

For example, a 2x2 identity matrix looks like \( I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \). The identity matrix is used in matrix algebra to preserve the value of a matrix when it is multiplied, and it also serves as the matrix equivalent of the multiplicative identity in a set of matrices.

Understanding the identity matrix is essential for grasping the nature of matrix inverses and for proving various properties about linear transformations in higher dimensions.

Matrix algebra encompasses a collection of mathematical operations that can be performed on matrices, including addition, subtraction, multiplication, and finding inverses. It is a fundamental concept used in various fields such as physics, economics, engineering, and computer science.

When performing operations like addition or subtraction, matrices involved must have the same dimensions. However, for multiplication, the number of columns in the first matrix must match the number of rows in the second matrix. And while matrix multiplication is associative and distributive, it is not commutative—meaning \( AB \) does not necessarily equal \( BA \) for matrices \( A \) and \( B \).

Furthermore, the concept of the determinant and inverse of a matrix plays a central role in matrix algebra, especially when solving linear systems or determining matrix properties. Understanding these core operations and properties is crucial for anyone studying linear algebra or related fields.