91Ó°ÊÓ

Matrix equations are a compact way to represent and solve linear systems of equations. For example, the system of equations given in the problem:\[ \begin{aligned} x_{1}+x_{2} &=4 \ 2 x_{1}-x_{2} &=14 \end{aligned} \]is transformed into a matrix equation: \[\begin{bmatrix} 1 & 1 \ 2 & -1 \end{bmatrix} \begin{bmatrix} x_{1} \ x_{2} \end{bmatrix} = \begin{bmatrix} 4 \ 14 \end{bmatrix}\]Here:- The **coefficient matrix** is the array \(A = \begin{bmatrix} 1 & 1 \ 2 & -1 \end{bmatrix} \).- The **variable matrix** \(X = \begin{bmatrix} x_{1} \ x_{2} \end{bmatrix} \) contains the unknowns.- The **constant matrix** \(B = \begin{bmatrix} 4 \ 14 \end{bmatrix} \) represents the constants from the original equations.The goal is to solve for \(X\) by manipulating the equation \(AX = B\). Understanding how matrix multiplication works is key to solving these equations effectively.

A 2x2 matrix is a simple square matrix that contains exactly two rows and two columns. It is often used in handling systems of two linear equations. For instance, in the given exercise, we have:\[ A = \begin{bmatrix} 1 & 1 \ 2 & -1 \end{bmatrix} \]Key points to remember about a 2x2 matrix include:- Two rows and two columns- Four elements in total- The position of elements matters significantly for calculationThis type of matrix can be inverted (if it is non-singular) which means we can find another matrix, called the inverse, that will help solve systems of equations using the equation \(A^{-1}AX = A^{-1}B = X\). Understanding these basics makes it easier to work with matrices to solve systems of equations.

The determinant of a matrix is a special value that can tell us if the matrix is invertible. For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated as:\[\det(A) = ad - bc\]In our example, with matrix \(A = \begin{bmatrix} 1 & 1 \ 2 & -1 \end{bmatrix}\), the determinant is:\[\det(A) = (1)(-1) - (1)(2) = -3\]Why is the determinant important?- It tells us whether a matrix is invertible (non-zero determinant)- Influences the formula for the inverse of a 2x2 matrixIf \(\det(A)\) is zero, the matrix does not have an inverse and cannot be used to solve systems of equations via matrix methods. Thus, calculating the determinant is a critical step in confirming that a solution path exists.

Solving a system of linear equations with matrices relies heavily on understanding and applying inverse matrices. The primary equation used is:\[ AX = B \]where \(A\) is a known matrix, \(X\) is the matrix of variables, and \(B\) is the matrix of constants. To find \(X\), we use:\[ X = A^{-1}B \]Let’s look at an example from our solution:- As found, \(A^{-1} = \begin{bmatrix} \frac{1}{3} & \frac{1}{3} \ \frac{2}{3} & -\frac{1}{3} \end{bmatrix} \)- The multiplication \(A^{-1}B = \begin{bmatrix} 6 \ -2 \end{bmatrix} \)This means the values of the variables are \(x_1 = 6\) and \(x_2 = -2\). Understanding matrix operations simplifies complex equations and provides an efficient method for finding solutions.