ILAM - Use Matrices to Solve Systems Lesson

Use Matrices to Solve Systems

One of the most common applications of matrices is to use them to solve systems of equations. Remember, you learned how to solve systems by graphing, substitution and elimination! 

Let's say we have this system of equations:

$2x+9y=-1\\ 4x+y=15$

We can rewrite that as a matrix equation:

$\begin{bmatrix} 2 & 9 \\ 4 & 1 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} -1 \\ 15 \end{bmatrix}$

Then, we can use the inverse of the coefficient matrix to isolate the variables. Consider this matrix equation below:

$A\cdot X=B\\ A^{-1}\cdot AX=A^{-1}\cdot B\\ 1X=A^{-1}\cdot B$

So, what I am doing is multiplying by the inverse matrix on both sides to create the Identity Matrix, which is like a 1 so that I can consider the variable matrix alone! Remember matrix multiplication is not commutative - so we must multiply matrix $A^{-1}\:by\:B,\:and\:not\:B\:by\:A^{-1}$. Therefore, You must multiply inverse A with B. Not B x inverse A. Watch this video to see how to solve the equation above by hand and with a calculator.

 

You might be given a 3x3 matrix! We will strictly use technology to solve this type of equation. Watch this video:

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