The inverse matrix of \(\left[\begin{array}{ll}2 & 3 \\ 1 & 1\end{array}\right]\) is a) \(\left[\begin{array}{rr}-1 & 3 \\ 1 & -2\end{array}\right]\) c) \(\left[\begin{array}{ll}-1 & 1 \\ -3 & 2\end{array}\right]\) b) \(\left[\begin{array}{rr}1 & -1 \\ -3 & 2\end{array}\right]\) d) \(\left[\begin{array}{rr}-2 & 3 \\ 1 & -1\end{array}\right]\)

Understanding the determinant of a matrix is essential in linear algebra because it has significant implications for matrix properties. Essentially, the determinant can be seen as a scalar value that encodes certain information about a matrix, such as whether it's invertible or not, and if so, how volume might change under the linear transformation the matrix represents.

For a 2x2 matrix, the formula for the determinant is relatively straightforward. If you consider a matrix \( A = \left[\begin{array}{cc}a & b \ c & d \end{array}\right] \), its determinant is calculated as \( \mathrm{det}(A) = ad - bc \). This value can be positive, negative, or zero. A nonzero determinant implies that the matrix has an inverse, and the associated linear transformation is invertible (meaning it doesn't compress space into a lower dimension). In contrast, a determinant of zero indicates that the matrix is 'singular' and does not have an inverse.

The adjugate, or adjoint, of a matrix plays a pivotal role when it comes to finding the inverse of a matrix. It's another matrix that's derived from the original matrix by taking the transposition of its cofactor matrix.

For a 2x2 matrix \( A \), the procedure to calculate its adjugate \( \mathrm{adj}(A) \) is relatively simple. You switch the positions of the elements in the leading diagonal, that is, swap \( a \) and \( d \) in our matrix example. Then, change the sign of the other two elements, \( b \) and \( c \). The resulting matrix is then the adjugate of \( A \).

So if you have \( A = \left[\begin{array}{cc}a & b \ c & d \end{array}\right] \), the adjugate of A, \( \mathrm{adj}(A) \) would be \( \left[\begin{array}{cc}d & -b \ -c & a \end{array}\right] \). The adjugate has the property that when it's multiplied by the original matrix, it yields the product of the determinant of the original matrix and the identity matrix.

Matrix inversion is a method to find the matrix (denoted as \( A^{-1} \) for a given matrix \( A \)) that, when multiplied with the original matrix, yields the identity matrix. However, it is essential to note that not every matrix has an inverse; only those with a nonzero determinant do.

To invert a matrix, one must divide the adjugate of the matrix by its determinant: \( A^{-1} = \frac{1}{\mathrm{det}(A)} \times \mathrm{adj}(A) \). The process for a 2x2 matrix involves the steps covered in our textbook solution—calculating the determinant, finding the adjugate, and then combining these to form the inverse matrix. In our case, with \( \mathrm{det}(A) = -1 \) and \( \mathrm{adj}(A) \), the inverse matrix is indeed \( \left[\begin{array}{rr} -1 & 3 \ 1 & -2 \end{array}\right] \), corresponding to option (a).

This process is central to numerous applications in linear algebra, like solving systems of linear equations, and is also fundamental in more advanced areas like computer graphics and optimization.

Linear algebra is a branch of mathematics that deals with vector spaces and the linear mappings between them. It's the area of math that explores lines, planes, and their relationships through algebraic equations and concepts. Linear algebra is foundational for advanced mathematics but also for many applications in science and engineering.

Key concepts in linear algebra include vectors, matrices, determinants, and linear transformations. Matrices can represent linear transformations, and the determinant of a matrix provides insight into various properties of these transformations, like whether they are invertible. As we've seen in the earlier sections, calculation of the inverse matrix requires both the determinant and the adjugate, showcasing how these concepts interplay within linear algebra.

Moreover, linear algebra is not just about calculations. It also provides a language and a framework for solving more complex problems, like those found in physics, computer science, and machine learning. Understanding the concepts behind things like matrix inversion can not only help solve textbook exercises but also develop a deeper comprehension of the principles that govern various scientific domains.