Solving equations with inverse matrices
Let’s suppose the equation is: [Tex]a_1x+a_1y+a_3z=d_1 \\b_1x+b_2y+b_3z=d_2\\ c_1x+c_2y+c_3z=d_3[/Tex]
Matrix Method is used to find the solution of the system of the equations. In the equations, all of the variables should be written in the proper order. On the appropriate sides, write the variables, their coefficients, and constants.
The method of determining the inverse is used to solve a system of linear equations, and it requires two additional matrices. The variables are represented by Matrix X. The constants are represented by Matrix B. Using matrix multiplication, a system of equations with the same number of equations as a variable is defined as,
AX=B
Let A be the coefficient matrix, X be the variable matrix, and B be the constant matrix to solve a system of linear equations with an inverse matrix. As a result, we’d want to solve the system AX = B. Take a look at the equations below as an example.
[Tex]\left[\begin{matrix}a_1x+a_2y+a_3z\\b_1x+b_2y+b_3z\\c_1x+c_2y+c_3z\\\end{matrix}\right]=\left[\begin{matrix}d_1\\d_2\\d_3\\\end{matrix}\right][/Tex]
[Tex]\left[\begin{matrix}a_1&a_2&a_3\\b_1&b_2&b_3\\c_1&c_2&c_3\\\end{matrix}\right]\left[\begin{matrix}x\\y\\z\\\end{matrix}\right]=\left[\begin{matrix}d_1\\d_2\\d_3\\\end{matrix}\right][/Tex]
AX = B
where: [Tex]A=\left[\begin{matrix}a_1&a_2&a_3\\b_1&b_2&b_3\\c_1&c_2&c_3\\\end{matrix}\right],X=\left[\begin{matrix}x\\y\\z\\\end{matrix}\right],B=\left[\begin{matrix}d_1\\d_2\\d_3\\\end{matrix}\right][/Tex]
Case 1: If A is a nonsingular matrix, it has an inverse.
Let A be the coefficient matrix, X be the variable matrix, and B be the constant matrix to solve a system of linear equations with an inverse matrix. As a result, we’d want to solve the system AX=B. To get the answer, multiply both sides by the inverse of A.
[Tex](A^{-1})AX=(A^{-1})B [(A^{-1})A]X=(A^{-1})B IX=(A^{-1})B X=(A^{-1})B[/Tex]
As the inverse of a matrix is unique, this matrix equation offers a unique solution to the given system of equations. The Matrix Method is a method for solving systems of equations.
Case 2: If A is a singular matrix, then | A| = 0. In this case, calculate (adj A) B.
If (adj A) B ≠ O, (O being zero matrices), then the solution does not exist and the system of equations is called inconsistent.
If (adj A) B = O, then the system may be either consistent or inconsistent accordingly as the system has either infinitely many solutions or no solution.
How to Solve a System of Equations using Inverse of Matrices?
How to Solve a System of Equations Using Inverse of Matrices? In mathematics, a matrix is an array of numbers arranged in a rectangular pattern and separated into rows and columns. They’re commonly depicted by enclosing all of the integers within square brackets.
In this article, we will discuss how to solve a system of equations using the inverse of matrices in detail.
Table of Content
- Determinant
- Minors and Cofactors
- Adjoint of a matrix
- Inverse of a matrix
- Application of Matrices and Determinants
- Representing linear systems with matrix equations
- Solving equations with inverse matrices
- Problems on How to Solve a System of Equations using Inverse of Matrices?
- Practice Problems on How to Solve a System of Equations using Inverse of Matrices?