Matrix Methods Calculator

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Example

Created on 2024-06-20Asked by Ethan Baker (Solvelet student)

Solve the system of linear equations using matrix methods:

\begin{cases} x + 2y = 5 \\ 3x + 4y = 6 \end{cases}

Solution

To solve the system of linear equations using matrix methods:

\begin{cases} x + 2y = 5 \\ 3x + 4y = 6 \end{cases}

we represent the system as a matrix equation

AX = B

: 1. Write the coefficient matrix

A

, variable matrix

X

, and constant matrix

B

A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \end{pmatrix}, \quad B = \begin{pmatrix} 5 \\ 6 \end{pmatrix}

2. The matrix equation is:

\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ 6 \end{pmatrix}

3. Solve for

X

by finding the inverse of

A

and multiplying by

B

X = A^{-1} B

4. Compute the inverse of

A

A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \frac{1}{-2} \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix} = \begin{pmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{pmatrix}

5. Multiply

A^{-1}

B

X = \begin{pmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{pmatrix} \begin{pmatrix} 5 \\ 6 \end{pmatrix} = \begin{pmatrix} (-2)(5) + (1)(6) \\ (\frac{3}{2})(5) + (-\frac{1}{2})(6) \end{pmatrix} = \begin{pmatrix} -10 + 6 \\ \frac{15}{2} - \frac{6}{2} \end{pmatrix} = \begin{pmatrix} -4 \\ \frac{9}{2} \end{pmatrix}

Therefore, the solution to the system is:

x = -4, \quad y = \frac{9}{2}

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Noah Brown on Solvelet

1. Solve the system of equations

AX = B

using the LU decomposition method, where

A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}

and

B = \begin{bmatrix} 5 \\ 7 \end{bmatrix}

.2. Perform the QR factorization of the matrix

C = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}

DefinitionMethods used in matrices are matrix methods to solve linear equations, perform operations on 2D sequences or create mathematical models. The methods consist in linear algebra functions as Gaussian elimination, matrix inversion, and eigenvalue decomposition. For example gaussian elimination for the equations represented by Ax=b.

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