The calculator will find the inverse (if it exists) of the square matrix using the Gaussian elimination method or the adjugate method, with steps shown. Show Related calculators: Gauss-Jordan Elimination Calculator, Pseudoinverse Calculator Your InputCalculate $$$\left[\begin{array}{cc}2 & 1\\1 & 3\end{array}\right]^{-1}$$$ using the Gauss-Jordan elimination. SolutionTo find the inverse matrix, augment it with the identity matrix and perform row operations trying to make the identity matrix to the left. Then to the right will be the inverse matrix. So, augment the matrix with the identity matrix: $$$\left[\begin{array}{cc|cc}2 & 1 & 1 & 0\\1 & 3 & 0 & 1\end{array}\right]$$$ Divide row $$$1$$$ by $$$2$$$: $$$R_{1} = \frac{R_{1}}{2}$$$. $$$\left[\begin{array}{cc|cc}1 & \frac{1}{2} & \frac{1}{2} & 0\\1 & 3 & 0 & 1\end{array}\right]$$$ Subtract row $$$1$$$ from row $$$2$$$: $$$R_{2} = R_{2} - R_{1}$$$. $$$\left[\begin{array}{cc|cc}1 & \frac{1}{2} & \frac{1}{2} & 0\\0 & \frac{5}{2} & - \frac{1}{2} & 1\end{array}\right]$$$ Multiply row $$$2$$$ by $$$\frac{2}{5}$$$: $$$R_{2} = \frac{2 R_{2}}{5}$$$. $$$\left[\begin{array}{cc|cc}1 & \frac{1}{2} & \frac{1}{2} & 0\\0 & 1 & - \frac{1}{5} & \frac{2}{5}\end{array}\right]$$$ Subtract row $$$2$$$ multiplied by $$$\frac{1}{2}$$$ from row $$$1$$$: $$$R_{1} = R_{1} - \frac{R_{2}}{2}$$$. $$$\left[\begin{array}{cc|cc}1 & 0 & \frac{3}{5} & - \frac{1}{5}\\0 & 1 & - \frac{1}{5} & \frac{2}{5}\end{array}\right]$$$ We are done. On the left is the identity matrix. On the right is the inverse matrix. AnswerThe inverse matrix is $$$\left[\begin{array}{cc}\frac{3}{5} & - \frac{1}{5}\\- \frac{1}{5} & \frac{2}{5}\end{array}\right] = \left[\begin{array}{cc}0.6 & -0.2\\-0.2 & 0.4\end{array}\right].$$$A
Solve the system of linear equations step by stepThis calculator will solve the system of linear equations of any kind, with steps shown, using either the Gauss-Jordan elimination method, the inverse matrix method, or Cramer's rule. Related calculator: System of Equations Calculator Comma-separated, for example, x+2y=5,3x+5y=14. Leave empty for autodetection or specify variables like x,y (comma-separated). If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below. Your InputSolve $$$\begin{cases} 5 x - 2 y = 1 \\ x + 3 y = 7 \end{cases}$$$ for $$$x$$$, $$$y$$$ using the Gauss-Jordan Elimination method. SolutionWrite down the augmented matrix: $$$\left[\begin{array}{cc|c}5 & -2 & 1\\1 & 3 & 7\end{array}\right]$$$. Perform the Gauss-Jordan elimination (for steps, see Gauss-Jordan elimination calculator): $$$\left[\begin{array}{cc|c}5 & -2 & 1\\0 & \frac{17}{5} & \frac{34}{5}\end{array}\right]$$$. Back-substitute: $$$y = \frac{\frac{34}{5}}{\frac{17}{5}} = 2$$$ $$$x = \frac{1 - \left(-2\right) \left(2\right)}{5} = 1$$$ Answer$$$x = 1$$$A $$$y = 2$$$A This calculator solves Systems of Linear Equations using Gaussian Elimination Method, Inverse Matrix Method, or Cramer's rule. Also you can compute a number of solutions in a system of linear equations (analyse the compatibility) using Rouché–Capelli theorem. Enter coefficients of your system into the input fields. Leave cells empty for variables, which do not participate in your equations. To input
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How do you solve a system of equations using an inverse matrix?SOLVING A SYSTEM OF EQUATIONS USING THE INVERSE OF A MATRIX. Given a system of equations, write the coefficient matrix A, the variable matrix X, and the constant matrix B. Then.. Multiply both sides by the inverse of A to obtain the solution.. What is the inverse matrix method?The concept of inverse of a matrix is a multidimensional generalization of the concept of reciprocal of a number: the product between a number and its reciprocal is equal to 1; the product between a square matrix and its inverse is equal to the identity matrix.
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