91Ó°ÊÓ

The inverse of a matrix is a fundamental concept when solving systems of linear equations. The inverse of a matrix, often denoted as \(A^{-1}\), is a matrix that, when multiplied with the original matrix, yields the identity matrix. This identity matrix acts like the number 1 in ordinary algebra.
For a 2x2 matrix \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\), the inverse is calculated using the formula:
\[A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}\]. The term \(ad - bc\), called the determinant, must be non-zero for the inverse to exist. If the determinant is zero, the matrix does not have an inverse and the system cannot be solved using this method.
In our exercise, the coefficient matrix \(A = \begin{pmatrix} 1 & 3 \ 2 & -1 \end{pmatrix}\). Its determinant is \(1(-1) - 3(2) = -7\), which is non-zero. Thus, we can find the inverse matrix A^{-1}.

Matrix multiplication is a key operation used in solving systems of linear equations, among many other applications. When multiplying two matrices, the number of columns in the first matrix must match the number of rows in the second matrix.
To multiply matrix \(A\) by matrix \(B\), you follow these steps:

• Multiply elements of the rows of \(A\) by elements of the columns of \(B\).
• Sum these products to get the corresponding element in the resultant matrix.

For example, if \(A_{m \times n}\) is multiplied by \(B_{n \times p}\), the result is a matrix \(C_{m \times p}\). In the exercise, to solve for matrix \(X\), we multiply \(A^{-1} \) (the inverse matrix) by \(B\) (the constants matrix). The multiplication of the 2x2 matrix by the 2x1 matrix gives us the 2x1 solution matrix \(X\).

The coefficient matrix is a matrix consisting of the coefficients of the variables in a system of linear equations. For a set of linear equations, you extract the numerical coefficients and arrange them into a matrix form.
For instance, in the system:
\[x + 3y = -12\ \ 2x - y = 11\]
The coefficient matrix is:
\[A = \begin{pmatrix} 1 & 3 \ 2 & -1 \end{pmatrix}\].
This matrix captures the relationship between variables and their coefficients, allowing us to rewrite the system in matrix form as \(AX = B\), where \(X\) is the variables matrix and \(B\) is the constants matrix. Solving the system involves manipulating these matrices to isolate the variables matrix \(X\). In our exercise, finding the inverse of the coefficient matrix allows us to solve for \(X\).