91Ó°ÊÓ

Matrix equations are a concise way to represent and solve systems of linear equations. Instead of dealing with multiple equations individually, we convert the system into a single matrix equation of the form: \(A\mathbf{x} = \mathbf{b}\). Here, \(A\) is the coefficients matrix, \(\mathbf{x}\) is the variable matrix, and \(\mathbf{b}\) is the constants matrix.
For example, consider the system: \(x + 3y = 16\) and \(6x + y = 11\). This can be written in matrix form as:
\[ \begin{pmatrix}1 & 3\ 6 & 1\end{pmatrix}\begin{pmatrix} x\ y\end{pmatrix} = \begin{pmatrix} 16\ 11\end{pmatrix} \]
Matrix equations simplify the process of solving complex systems by utilizing operations like matrix multiplication and finding inverses.

The inverse of a matrix is a crucial tool for solving matrix equations. If \(A\) is a square matrix, its inverse \(A^{-1}\) is defined such that \(A \cdot A^{-1} = I\), where \(I\) is the identity matrix. Not all matrices have inverses; a matrix must be non-singular (i.e., its determinant must not be zero).
For a 2x2 matrix \( \begin{pmatrix}a & b\ c & d\end{pmatrix} \), the inverse is:
\[ A^{-1} = \frac{1}{ad - bc}\begin{pmatrix}d & -b\ -c & a\end{pmatrix} \]
In our example, where \(A = \begin{pmatrix}1 & 3\ 6 & 1\end{pmatrix}\), we calculate the determinant as: \( \text{det}(A) = 1 \cdot 1 - 3 \cdot 6 = -17 \). Then, the inverse is:
\[ A^{-1} = \frac{1}{-17}\begin{pmatrix}1 & -3\ -6 & 1\end{pmatrix} = \begin{pmatrix} -\frac{1}{17} & \frac{3}{17}\ \frac{6}{17} & -\frac{1}{17}\end{pmatrix} \]

The determinant of a matrix is a special number that can provide lots of information about the matrix, including whether it has an inverse. For a 2x2 matrix \( \begin{pmatrix}a & b\ c & d\end{pmatrix} \), the determinant is calculated as: \( ad - bc \).
In the exercise, to find the inverse of matrix \(A = \begin{pmatrix}1 & 3\ 6 & 1\end{pmatrix} \), first, we calculate its determinant:
\[ \text{det}(A) = 1 \cdot 1 - 3 \cdot 6 = 1 - 18 = -17 \]
A non-zero determinant (-17 in this case) confirms that the matrix is invertible. Understanding how to calculate the determinant helps in determining the feasibility of finding an inverse, which is essential in solving systems of equations using matrices.

Matrix multiplication involves combining rows and columns from two matrices to produce a new matrix. This is especially important when solving matrix equations and working with inverses.
In the context of our example, once we find the inverse matrices, we perform matrix multiplication to solve for \( \mathbf{x} \). We multiply \(A^{-1} \mathbf{b}\) to find the variables \( x \) and \( y \):

\( \begin{pmatrix} x \ y\end{pmatrix} = \begin{pmatrix} -\frac{1}{17} & \frac{3}{17}\ \frac{6}{17} & -\frac{1}{17} \end{pmatrix} \begin{pmatrix} 16 \ 11 \end{pmatrix} \)

Carefully performing these multiplications:
\( x = -\frac{1}{17} \cdot 16 + \frac{3}{17} \cdot 11 = \frac{-16 + 33}{17} = 1 \)
\( y = \frac{6}{17} \cdot 16 - \frac{1}{17} \cdot 11 = \frac{96 - 11}{17} = 5 \)

Matrix multiplication allows us to solve for variables efficiently once we have the inverse matrix and constants matrix.