91Ó°ÊÓ

A matrix equation is a compact way of expressing a system of linear equations. In this exercise, we convert a system of three linear equations with three variables into the matrix equation format. This involves arranging the coefficients of variables in a square matrix, known as the coefficient matrix, denoted by \( A \). Similarly, the variables form a column matrix, \( \mathbf{x} \), and the constants from each equation form another column matrix, \( \mathbf{b} \). Thus, the system of equations:

• 4x - 10y + 3z = -19
• 2x + 5y - z = -7
• x - 3y - 2z = -2

can be rewritten as:\[ A\mathbf{x} = \mathbf{b} \] where: \[ A = \begin{bmatrix} 4 & -10 & 3 \ 2 & 5 & -1 \ 1 & -3 & -2 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \ y \ z \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} -19 \ -7 \ -2 \end{bmatrix} \] Each element of matrix \( A \) corresponds to a coefficient in the system, \( \mathbf{x} \) holds the variables, and \( \mathbf{b} \) represents the constants.

An inverse matrix is crucial when solving matrix equations. Specifically, for a matrix equation of the form \( A\mathbf{x} = \mathbf{b} \), the solution for \( \mathbf{x} \) can be found by calculating the inverse of the coefficient matrix \( A \), provided it exists. The inverse matrix \( A^{-1} \) must satisfy \( A A^{-1} = I \), where \( I \) is the identity matrix.When \( A \) is a 3x3 matrix, calculating the inverse involves:

• Finding the determinant of \( A \)
• Calculating the adjugate of \( A \)
• Using the formula: \( A^{-1} = \frac{1}{{\text{det}(A)}} \times \text{adj}(A) \)

Not every matrix has an inverse. If the determinant is zero, the matrix is non-invertible and the system of equations might not have a unique solution. In this context, you'd typically use tools like a scientific calculator or software to help compute the inverse effectively.

Matrix multiplication is a method used to solve the equation \( \mathbf{x} = A^{-1}\mathbf{b} \) once the inverse matrix \( A^{-1} \) is determined. This operation involves multiplying each element of the rows of \( A^{-1} \) by the corresponding elements of the column in \( \mathbf{b} \), summing these products to give a single number in the result matrix, \( \mathbf{x} \). For each row of \( A^{-1} \), this process is repeated to fill out the entire solution matrix:\( \begin{bmatrix} x \ y \ z \end{bmatrix} \).Matrix multiplication is not a straightforward term-by-term process; it follows specific rules:

• The number of columns in \( A^{-1} \) must equal the number of rows in \( \mathbf{b} \).
• Each element of the resulting matrix \( \mathbf{x} \) is calculated by multiplying the corresponding elements and summing the products.

The calculated matrix \( \mathbf{x} \) contains the values for \( x, y, \) and \( z \), which must be verified by substituting back into the original equations to confirm the solution's accuracy.