Solve matrix equations using inverses

Learning Objectives Represent a system of linear equations as a matrix equation of the form AX = B. Calculate the determinant of 2x2 and 3x3 matrices to determine if an inverse exists. Find the inverse of a 2x2 matrix using the standard formula. Find the inverse of a 3x3 matrix using the adjoint method. Solve matrix equations of the form AX = B by pre-multiplying by the inverse matrix A⁻¹. Verify the solution of a system by substituting the values back into the original equations. How can we solve a system of 10 equations with 10 variables without endless substitution? 🕵️‍♀️ Matrices provide an elegant and powerful method to do just that! This tutorial will guide you through a systematic process for solving systems of linear equations using matrix inverses. You will learn h...

TermDefinitionExample Matrix EquationAn equation in which a variable stands for a matrix. The most common form for solving systems of linear equations is AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.The system 2x + 3y = 7 and 4x - y = 3 can be written as the matrix equation [[2, 3], [4, -1]] * [[x], [y]] = [[7], [3]]. Identity Matrix (I)A square matrix with 1s on the main diagonal (from top-left to bottom-right) and 0s everywhere else. It is the multiplicative identity for matrices, meaning AI = IA = A.The 2x2 identity matrix is I₂ = [[1, 0], [0, 1]]. The 3x3 identity matrix is I₃ = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]. Inverse Matrix (A⁻¹)For a square matrix A, its inverse A⁻¹ is a unique matrix such that their product is the identity mat...

Solution to a Matrix Equation If AX = B, then X = A⁻¹B To solve for the variable matrix X, pre-multiply both sides of the equation by the inverse of the coefficient matrix A. The order of multiplication is critical. Inverse of a 2x2 Matrix For A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} This formula is used to find the inverse of a 2x2 matrix, provided the determinant (ad-bc) is not zero. You swap the elements on the main diagonal, negate the elements on the other diagonal, and multiply the resulting matrix by 1/determinant. Inverse of a 3x3 Matrix (Adjoint Method) A^{-1} = \frac{1}{\det(A)} \text{adj}(A) The inverse of a 3x3 (or larger) matrix is the adjoint of the matrix d...