## 17Calculus Precalculus - Solving Linear Systems Using Inverse Matrices

##### 17Calculus

There are three main techniques for solving linear systems of equations using matrices, row reduction, inverses and Cramer's Rule. Row reduction, which is the easiest and doesn't require a lot of knowledge about matrices, is covered on a separate page. Using inverses to solve linear systems of equations is discussed on this page. Finally, Cramer's Rule can also be used to solve linear systems of equations.

In order to solve linear systems using matrices, first we set up the matrices, then find the inverse of the coefficient matrix and then do matrix multiplication. Here are the details.

Set Up The Matrices

To demonstrate this, we will use the following generic system.
$$\displaystyle{ \begin{array}{rcccl} ax & + & by & = & t \\ cx & + & dy & = & s \end{array} }$$

The matrices we need are
1. The coefficient matrix $$\displaystyle{ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} }$$
2. The variable matrix $$\displaystyle{ X = \begin{bmatrix} x \\ y \end{bmatrix} }$$
3. The constant matrix $$\displaystyle{ B = \begin{bmatrix} t \\ s \end{bmatrix} }$$

The linear system can now be written in matrix form as $$\displaystyle{ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} t \\ s \end{bmatrix} }$$
or in shorthand notation $$AX=B$$

Preview of The Next Steps

Before we go into the details of the next steps, here is an idea of what we will be doing and why.
The idea is to find the inverse of matrix A, if it exists. Then we multiply both sides of the equation ON THE LEFT by the inverse. (Remember that matrix multiplication is not commutative.) So, if we denote the inverse of A as $$A^{-1}$$, then we have
$$\displaystyle{ \begin{array}{rcl} AX & = & B \\ A^{-1}AX & = & A^{-1}B \\ X & = & A^{-1}B \end{array} }$$
In the last step, we used the idea that $$A^{-1}A = I$$, where $$I$$ is the identity matrix, which works like multiplying by one, so $$IX = X$$. From the last equation, we can see that if we can find an inverse of matrix A, then all we need to do is multiply $$A^{-1}$$ ON THE LEFT of the constant matrix B to get the values of x and y, i.e. to solve the system.

Find The Inverse Of The Coefficient Matrix

Once you have set up the matrices from the given linear system, you need to find the inverse of the coefficient matrix. In our example, the coefficient matrix is
$$\displaystyle{ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} }$$
and this is the matrix we need to find the inverse of. For 2x2 matrices, the inverse is easy to find. For larger matrices, there are several techniques you can use. [See the inverse matrices page for details.]
Note - If the inverse matrix does not exist, this technique cannot be used.

Apply The Inverse Matrix

As mentioned above, we take the inverse matrix and multiply both sides of the equation $$AX=B$$ on the LEFT SIDE. This is critical. Matrix multiplication is not commutative. We need to multiply on the left of A and we need to do the same with B. So our equation looks like $$A^{-1}AX=A^{-1}B$$. Now, $$A^{-1}A = I$$, where $$I$$ is the identity matrix, so $$IX = X$$ and we are left with $$X=A^{-1}B$$, which is our final answer.

Okay, time for some practice problems.

Practice - 2x2 Systems

Unless otherwise instructed, solve the linear systems using inverse matrices, giving your answers in exact form.

$$3x+5y=-1;$$ $$x-2y=4$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$3x+5y=-1;$$ $$x-2y=4$$

$$x=18/11; ~ y=-13/11$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$3x+5y=-1;$$ $$x-2y=4$$

Solution

### PatrickJMT - 2005 video solution

video by PatrickJMT

$$x=18/11; ~ y=-13/11$$

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$$-x-3y=-2;$$ $$4x+5y=8$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$-x-3y=-2;$$ $$4x+5y=8$$

$$x=2; ~ y=0$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$-x-3y=-2;$$ $$4x+5y=8$$

Solution

### PatrickJMT - 2006 video solution

video by PatrickJMT

$$x=2; ~ y=0$$

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$$2x+5y=3;$$ $$x+3y=2$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$2x+5y=3;$$ $$x+3y=2$$

$$x=-1;~y=1$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$2x+5y=3;$$ $$x+3y=2$$

Solution

### Michel vanBiezen - 2007 video solution

video by Michel vanBiezen

$$x=-1;~y=1$$

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$$5x+y=8;$$ $$3x-4y=14$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$5x+y=8;$$ $$3x-4y=14$$

$$x=2;~y=-2$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$5x+y=8;$$ $$3x-4y=14$$

Solution

### Michel vanBiezen - 2008 video solution

video by Michel vanBiezen

$$x=2;~y=-2$$

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$$3x-y=5;$$ $$2x+y=5$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$3x-y=5;$$ $$2x+y=5$$

$$x=2;~y=1$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$3x-y=5;$$ $$2x+y=5$$

Solution

### MIP4U - 2010 video solution

video by MIP4U

$$x=2;~y=1$$

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$$3x+2y=7;$$ $$-6x+6y=6$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$3x+2y=7;$$ $$-6x+6y=6$$

$$x=1; ~ y=2$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$3x+2y=7;$$ $$-6x+6y=6$$

Solution

### Khan Academy - 2013 video solution

$$x=1; ~ y=2$$

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$$x-3y=-1;$$ $$4x+3y=11$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$x-3y=-1;$$ $$4x+3y=11$$

$$(2,1)$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$x-3y=-1;$$ $$4x+3y=11$$

Solution

### 2782 video solution

$$(2,1)$$

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$$5x + y = 8;$$ $$3x - 4y = 14$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$5x + y = 8;$$ $$3x - 4y = 14$$

$$(2,-2)$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$5x + y = 8;$$ $$3x - 4y = 14$$

Solution

### Michel vanBiezen - 2783 video solution

video by Michel vanBiezen

$$(2,-2)$$

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$$-x+y=1;$$ $$2x-y=1$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$-x+y=1;$$ $$2x-y=1$$

Solution

### 2784 video solution

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$$x-2y=-3;$$ $$3x+y=5$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$x-2y=-3;$$ $$3x+y=5$$

$$(1,2)$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$x-2y=-3;$$ $$3x+y=5$$

Solution

### 2785 video solution

$$(1,2)$$

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Practice - 3x3 Systems

$$x+2y+2z=5;$$ $$3x-2y+z=-6;$$ $$2x+y-z=-1$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$x+2y+2z=5;$$ $$3x-2y+z=-6;$$ $$2x+y-z=-1$$

$$x=-1; ~ y=2; ~ z=1$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$x+2y+2z=5;$$ $$3x-2y+z=-6;$$ $$2x+y-z=-1$$

Solution

### PatrickJMT - 2004 video solution

video by PatrickJMT

$$x=-1; ~ y=2; ~ z=1$$

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$$2x-3y-4z=2;$$ $$-z=5;$$ $$x-2y+z=3$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$2x-3y-4z=2;$$ $$-z=5;$$ $$x-2y+z=3$$

$$x=-60$$; $$y=-34$$; $$z=-5$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$2x-3y-4z=2;$$ $$-z=5;$$ $$x-2y+z=3$$

Solution

### Michel vanBiezen - 2009 video solution

video by Michel vanBiezen

$$x=-60$$; $$y=-34$$; $$z=-5$$

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$$x-3y+3z=-4;$$ $$2x+3y-z=15;$$ $$4x-3y-z=19$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$x-3y+3z=-4;$$ $$2x+3y-z=15;$$ $$4x-3y-z=19$$

$$x=5$$; $$y=1$$; $$z=-2$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$x-3y+3z=-4;$$ $$2x+3y-z=15;$$ $$4x-3y-z=19$$

Solution

### MIP4U - 2011 video solution

video by MIP4U

$$x=5$$; $$y=1$$; $$z=-2$$

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$$x+2y-z=7;$$ $$2x-3y-4z=-3;$$ $$x+y+z=0$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$x+2y-z=7;$$ $$2x-3y-4z=-3;$$ $$x+y+z=0$$

$$x=-1$$; $$y=3$$; $$z=-2$$

Problem Statement

Solve this linear system using inverse matrices, giving your answer in exact form.
$$x+2y-z=7;$$ $$2x-3y-4z=-3;$$ $$x+y+z=0$$

Solution

### MIP4U - 2012 video solution

video by MIP4U

$$x=-1$$; $$y=3$$; $$z=-2$$

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