91Ó°ÊÓ

In mathematics, an invertible matrix, also known as a non-singular or non-degenerate matrix, is a square matrix that has an inverse. A matrix is invertible if its determinant is non-zero. This means that there is another matrix, called the inverse, which when multiplied with the original matrix results in an identity matrix. The identity matrix is like the number 1 in multiplication for matrices; when you multiply a matrix by an identity matrix, you get the original matrix back.
Let's consider a matrix \( A \). If \( |A| eq 0 \), \( A \) is invertible. The formula to find the inverse of a 2x2 matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is given by:\[ A^{-1} = \frac{1}{|A|}\begin{pmatrix} d & -b \ -c & a \end{pmatrix} \]For larger matrices, though the concept stays the same, computing the inverse involves more complex methods, but the determinant still plays a crucial role. In our problem, it was key to understanding the invertibility of the matrices \( A \) and \( B \). Both being invertible means their determinants are not zero, supporting certain transformations and derivations we explored further.

Matrix transformations refer to operations such as rotating, scaling, translating, or performing linear transformations on a matrix. These transformations can alter properties of the matrix, such as its determinant. Common operations in transformations include row and column swaps, scalar multiplication, and addition of rows or columns.
Each of these actions affects the matrix determinant differently:

• Swapping two rows or two columns of a matrix will multiply the determinant by \(-1\).
• Multiplying a row or column by a constant results in the determinant being multiplied by that constant.
• Adding a multiple of one row to another row does not change the determinant.

In our exercise, matrix \( B \) was formed by performing certain row and element transformations on matrix \( A \), which changed the sign of its determinant. Understanding these properties helps in determining the correctness of various determinant-related statements, such as whether \(|A| = -|B|\). All these transformations highlight the delicate relationship between the structure of the matrix and its determinant.

The adjugate of a matrix, often referred to as the adjoint, is a key concept in understanding matrix inverses, especially when dealing with determinants. For any matrix \( A \), the adjugate \( \text{adj}(A) \) is the transpose of the cofactor matrix of \( A \). Each element of the cofactor matrix is the determinant of a minor of \( A \), multiplied by \(-1\) raised to the power of a sum of indices.The adjugate matrix is used in the formula for the inverse of matrices larger than 2x2, particularly when determining the inverse of \( A \) as:\[ A^{-1} = \frac{1}{|A|} \text{adj}(A) \]If \( A \) is invertible (as in our problem), then the determinant \(|A|\) is non-zero, and thus \( \text{adj}(A) \) contributes to the computation of \( A^{-1} \). The properties of adjugate matrices align closely with those of determinants, such that \(|\text{adj}(A)| = |A|^{n-1}\) for an \( n \times n \) matrix. This was relevant in assessing the given statement about the relationship between \( \text{adj}(A) \) and \( \text{adj}(B) \), and understanding whether they should be equal or not in magnitude. This understanding is essential in solving problems that deal with matrix determinants and inverses.