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Understanding the matrix equation concept is critical when solving systems of linear equations using matrices. If we translate the given system of linear equations into a matrix equation, we can express it compactly as AX = B. Here, A is called the coefficient matrix; it holds the coefficients of the variables from the system. The matrix X contains the variables 'x', 'y', and 'z', and the matrix B contains the constants from the right side of the equations.
For our given exercise, the coefficient matrix A is \[\begin{bmatrix}1 & -1 & 2 \3 & 4 & -5 \2 & -1 & 3\end{bmatrix}\], and the constant matrix B is \[\begin{bmatrix}7 \-5 \12\end{bmatrix}\], with the variable matrix X being \[\begin{bmatrix}x \y \z\end{bmatrix}\]. The matrix equation AX = B succinctly represents the given system, which can unfold into the original equations when multiplied out. The goal is to find X, such that the equation holds true.

When we have a matrix equation like AX = B, one approach to solve for X is to use the inverse of the coefficient matrix A. The inverse of a matrix A, denoted A鈦宦�, is a special matrix that, when multiplied by A, returns the identity matrix, thereby effectively 'canceling out' the effects of A. Finding the inverse of a matrix is like finding the reciprocal of a number鈥攎ultiplying the number by its reciprocal gives you 1.
For a square matrix to have an inverse, it must be non-singular, meaning it has a determinant that is not zero. Various methods, including the adjugate and Gauss-Jordan elimination methods, can be utilized to calculate the inverse. Once A鈦宦� is determined for our exercise, it is used to isolate X by multiplying both sides of the matrix equation by A鈦宦�, leading to A鈦宦笰X = A鈦宦笲. Since A鈦宦笰 is the identity matrix, this simplifies to X = A鈦宦笲, giving us a direct way to calculate the values for X.

Matrix multiplication is a key operation when solving systems of linear equations with matrices. Once we've obtained the inverse of the coefficient matrix (A鈦宦�) from our exercise, it's time to multiply A鈦宦� by the constant matrix B to find our variable matrix X. In matrix multiplication, we multiply the rows of the first matrix by the columns of the second matrix, summing the products. An important note is that matrix multiplication is not commutative, which means A鈦宦笲 is not necessarily the same as BA鈦宦�.
For our exercise, after computing A鈦宦�, we multiply it by B as follows: \[A^{-1} * B = X\].This will give us the matrix X, which contains the solved values of 'x', 'y', and 'z'. Remember that for the multiplication to be defined, the number of columns in A鈦宦� must match the number of rows in B. By meticulously executing the multiplication step, we ensure that our solution is calculated accurately, yielding the values of the variables that satisfy the system of equations.