91Ó°ÊÓ

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements. Matrices are fundamental to various calculations in mathematics and are used as a tool to represent and solve systems of linear equations. For example, the matrices A and B provided in the exercise are 2x2 matrices, meaning they each have two rows and two columns.

The notation of a matrix often involves large square brackets enclosing the elements, with each row of numbers separated by a new line. For example, matrix A from the exercise is represented as \[ \begin{array}{cc} 2 & -9 \ 1 & -4 \end{array} \], and similarly for B. Matrices can come in various sizes, and operations such as addition, subtraction, and multiplication can be performed on them, provided certain conditions are met in terms of their dimensions.

The inverse of a matrix A, denoted as A^-1, is a matrix that, when multiplied by A, results in the identity matrix. The identity matrix is a special kind of matrix that has 1's on its main diagonal and 0's elsewhere, with its effect being analogous to multiplying a number by 1 in regular arithmetic - the original value remains unchanged.

Not all matrices have an inverse. For a matrix to be invertible, it must be a square matrix (same number of rows and columns), and it must have a non-zero determinant. The determinant of a 2x2 matrix A is calculated as ad - bc, where a, b, c, d are the elements of the matrix, arranged as \[ \begin{array}{cc} a & b \ c & d \end{array} \]. The formula for the inverse of a 2x2 matrix, as seen in step 1 of the solution, involves placing the elements d and a in the opposite positions, negating the off-diagonal elements, and dividing by the determinant.

Matrix multiplication involves taking two matrices and producing another matrix by multiplying matching elements and summing their products. Importantly, this operation is not element-wise—each element in the resulting matrix is a sum of products. The number of columns in the first matrix must match the number of rows in the second matrix for matrix multiplication to be possible.

In the case of our 2x2 matrices A and B, to multiply them we would perform a series of multiplications and additions. For example, the element at the first row and first column of the resulting matrix would be a combination of multiplying the first element of A's first row with the first element of B's first column, and the second element of A's first row with the second element of B's first column, and then adding these products together. Matrix multiplication is associative but not commutative, meaning that while A(BA) is equal to (AB)A, AB is generally not equal to BA.

A 2x2 matrix is a square matrix that has two rows and two columns. Because of its small size, operations including computation of the determinant, inverse, and multiplication can be done relatively simply and with a specific set of formulas as shown in the initial solution. The formula for the inverse of a 2x2 matrix A is particulary straight-forward: \(A^{-1} = \frac{1}{{(a*d - b*c)}} * \left[ \begin{array}{cc} d & -b \ -c & a \end{array} \right]\), where a, b, c, d are the elements of the matrix, and ad - bc is the determinant of A.

Given their simplicity, 2x2 matrices are often used to introduce the concepts of matrix operations. It's also easy to visually demonstrate various matrix operations with 2x2 matrices, which aids in the foundational understanding of linear algebra concepts. In the context of the exercise, matrices A and B were 2x2 matrices, which allowed the quick application of the inverse formula.