311.2.2.6 use inverse matrix to solve system of equations

• MHB
• karush
In summary, an inverse matrix is a square matrix that can be multiplied by the original matrix to get the identity matrix. It is useful in solving systems of equations as it simplifies the process and allows for quick and efficient solutions. The inverse matrix can be found using the Gauss-Jordan elimination or adjugate matrix methods. However, not all matrices have an inverse and there are limitations to using it, such as the need for the same number of variables and equations and the possibility of computational complexity for larger matrices.
karush
Gold Member
MHB
$\tiny{311.2.2.6}$
Use the inverse to solve the system
$\begin{array}{rrrrr} 7x_1&+3x_2&=-9\\ -2x_1&+x_2&=10 \end{array}$

$A^{-1}b=x$
$\begin{bmatrix} \frac{1}{13}&-\frac{3}{13} \\ \\ \frac{2}{13}& \frac{7}{13} \end{bmatrix} \begin{bmatrix} -9\\ \\ 10 \end{bmatrix} = \begin{bmatrix} -3\\ \\ 4 \end{bmatrix}$

the thing I could not get here without a calculator is $A^{-1}$

Given $$\displaystyle A = \left ( \begin{matrix} a & b \\ c & d \end{matrix} \right )$$

When all else fails:
$$\displaystyle A^{-1} = \dfrac{1}{ |A| } \left ( \begin{matrix} d & -b \\ -c & a \end{matrix} \right )$$

I have this one memorized.

-Dan

Last edited by a moderator:
ok i think i had a and switched

1. What is an inverse matrix?

An inverse matrix is a matrix that when multiplied by the original matrix, results in the identity matrix. It is denoted by A-1 and is only possible for square matrices with a non-zero determinant.

2. How is an inverse matrix used to solve a system of equations?

To solve a system of equations using an inverse matrix, the system must first be written in matrix form. The coefficient matrix is then inverted, and the resulting inverse matrix is multiplied by the constant matrix. The solution can be found by equating the resulting matrix to the variable matrix.

3. When can an inverse matrix not be used to solve a system of equations?

An inverse matrix cannot be used to solve a system of equations if the coefficient matrix is not square or has a determinant of zero. In these cases, other methods such as Gaussian elimination or Cramer's rule must be used.

4. What is the importance of using an inverse matrix to solve a system of equations?

Using an inverse matrix to solve a system of equations is important because it provides an efficient and accurate method for finding the solution. It also allows for the use of matrix operations, which can be useful in solving larger and more complex systems of equations.

5. Can an inverse matrix be used for any type of system of equations?

No, an inverse matrix can only be used for systems of linear equations. It cannot be used for systems of non-linear equations, as the resulting matrix would not be an inverse of the coefficient matrix.

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