# 6. Matrices and Linear Equations

by M. Bourne

We wish to solve the system of simultaneous linear equations using matrices:

a_{1}x+b_{1}y=c_{1}

a_{2}x+b_{2}y=c_{2}

If we let

`A=((a_1,b_1),(a_2,b_2))`, `\ X=((x),(y))\ ` and `\ C=((c_1),(c_2))`

then `AX=C`*.* (We first saw this in Multiplication of Matrices).

If we now multiply each side of

AX=C

on the left by

A^{-1}, we have:

A^{-1}AX=A^{-1}C.

However, we know that *A*^{-1}*A* =
*I*, the Identity matrix. So we obtain

IX=A^{-1}C.

But *IX* = *X*, so the solution to the system of
equations is given by:

X=A^{-1}C

See the box at the top of Inverse of a Matrix for more explanation about why this works.

**Note:** We **cannot** reverse the order of multiplication and use *CA*^{-1} because
matrix multiplication is not commutative.

### Example - solving a system using the Inverse Matrix

Solve the system using matrices.

−

x+ 5y= 42

x+ 5y= −2

Always check your solutions!

Answer

We have:

`A=((-1,5),(2,5)),` ` \ X=((x),(y))\ ` and `\ C=((4),(-2))`

To solve the system, we need the inverse of *A*, which we write as *A*^{-1}.

Swap leading diagonal:

`((5,5),(2,-1))`

Change signs of the other 2 elements:

`((5,-5),(-2,-1))`

Now we find the determinant of *A*:

`|A| = -5 - 10 = -15`

So

`A^-1` `=-1/15((5,-5),(-2,-1))` ` = ((-1/3,1/3),(2/15,1/15))` ` = ((-0.333,0.333),(0.133,0.067)) `

So the solution to the system is given by:

`X=A^-1C` `=((-0.333,0.333),(0.133,0.067))((4),(-2))` `=((-2),(0.4)) `

This answer means that we have found the solution `x = -2` and `y = 0.4`.

Is the solution correct?

We check it in the original set of equations:

`{:(-x+5y,=4),(2x+5y,=-2):}`

Substituting `x = -2` and `y = 0.4`, we get:

`−(−2) + 5×(0.4) = 2 + 2 = 4` [Checks OK]

`2×(−2) + 5×(0.4)` ` = −4 + 2` ` = −2` [Checks OK]

So the solution to the original system of equations is

`x = -2,\ \ y = 0.4`.

## Solving 3×3 Systems of Equations

We can extend the above method to systems of any size. We cannot use the same method for finding inverses of matrices bigger than 2×2.

We will use a Computer Algebra System to find inverses larger than 2×2.

### Example - 3×3 System of Equations

Solve the system using matrix methods.

`{: (x+2y-z=6),(3x+5y-z=2),(-2x-y-2z=4) :}`

Did I mention? It's a good idea to always check your solutions.

Answer

`A=((1,2,-1),(3,5,-1),(-2,-1,-2)),` `X=((x),(y),(z)),` and `C=((6),(2),(4))`

Using Scientific Notebook, we find the inverse of *A* to be:

`A^-1=((5.5,-2.5,-1.5),(-4,2,1),(-3.5,1.5,0.5))`

(We could have used Gauss-Jordan Elimination if we need to show all steps.)

So the solution to the system of equations is:

`X=A^-1C`

`=((5.5,-2.5,-1.5),(-4,2,1),(-3.5,1.5,0.5))((6),(2),(4))`

`=((22),(-16),(-16))`

Check:

`22 + 2(-16) - (-16) = 6` [Checks OK]

`3(22) + 5(-16) - (-16) = 2` [Checks OK]

`-2(22) - (16) - 2(-16) = 4` [Checks OK]

So the solution is `x = 22`, `y = -16` and `z = -16`.

### Example - Electronics application of 3×3 System of Equations

Find the electric currents shown by solving the matrix equation (obtained using Kirchhoff's Law) arising from this circuit:

`((I_1+I_2+I_3),(-2I_1+3I_2),(-3I_2+6I_3))=((0),(24),(0))`

(You can explore what the solution for this example really means in this 3D interactive systems of equations applet.)

Answer

We can write this as:

`((1,1,1),(-2,3,0),(0,-3,6))((I_1),(I_2),(I_3))=((0),(24),(0))`

So we have:

`((I_1),(I_2),(I_3))=((1,1,1),(-2,3,0),(0,-3,6))^-1((0),(24),(0))`

Using a computer algebra system to perform the inverse and multiply by the constant matrix, we get:

`I_1= -6\ "A"`

`I_2= 4\ "A"`

`I_3= 2\ "A"`

We observe that *I*_{1} is negative, as expected from the circuit diagram.

### Exercise 1

The following equations are found in a particular electrical circuit. Find the currents using matrix methods.

`{: (I_A+I_B+I_C=0),(2I_A-5I_B=6),(5I_B-I_C=-3) :}`

(This example is also included in the 3D interactive systems of equations applet.)

Answer

We need to form the matrices:

`A=((1,1,1),(2,-5,0),(0,5,-1))`, `\ X=((I_A),(I_B),(I_C))\ ` and `\ C=((0),(6),(-3))`

Using Scientific Notebook (or any Computer Algebra System), we find:

`A^-1` ` = ((0.294,0.353,0.294),(0.118,-0.059,0.118),(0.588,-0.294,-0.412))`

and so (continuing to use Scientific Notebook, with rounding to 3 decimal places):

`((I_A),(I_B),(I_C))=A^-1C`

`=((0.294,0.353,0.294),(0.118,-0.059,0.118),(0.588,-0.294,-0.412))((0),(6),(-3))`

`=((1.236),(-0.708),(-0.528))`

Therefore

`I_A= 1.236\ "A"`,

`I_B= -0.708\ "A"` and

`I_C= -0.528\ "A"`

### Exercise 2

Recall this problem from before? If we know the simultaneous equations involved, we will be able to solve the system using inverse matrices on a computer.

The circuit equations, using Kirchhoff's Law:

−26 = 72

I_{1}− 17I_{3}− 35I_{4}34 = 122

I_{2}− 35I_{3}− 87I_{7}−4 = 233

I_{7}− 87I_{2}− 34I_{3}− 72I_{6}−13 = 149

I_{3}− 17I_{1}− 35I_{2}− 28I_{5}− 35I_{6}− 34I_{7}−27 = 105

I_{5}− 28I_{3}− 43I_{4}− 34I_{6}24 = 141

I_{6}− 35I_{3}− 34I_{5}− 72I_{7}5 = 105

I_{4}− 35I_{1}− 43I_{5}

What are the individual currents, *I*_{1} to *I*_{7}?

### Phone users

**NOTE:** If you're on a phone, you can scroll any **wide matrices** on this page to the right or left to see the whole expression.

Answer

#### Solving currents in a Circuit (7 × 7 system)

We solve this using a computer as follows. We just write the coefficient matrix on the left, find the inverse (raise the matrix to the power -1) and multiply the result by the constant matrix.

You can use Matlab, Mathcad or similar math software to do this. Wolfram|Alpha is a free alternative.

`X=[(72,0,-17,-35,0,0,0), (0,122,-35,0,0,0,-87), (0,-87,-34,0,0,-72,233), (-17,-35,149,0,-28,-35,-34), (0,0,-28,-43,105,-34,0), (0,0,-35,0,-34,141,-72), (-35,0,0,105,-43,0,0)]^-1 [(-26),(34),(-4),(-13),(-27),(24),(5)]`

`=[(-0.46801),(0.42932),(5.193xx10^-3),(-0.22243),(-0.27848),(0.21115),(0.20914)]`

The answer means that the currents in this circuit are (to 4 decimal places):

`I_1 = -0.4680\ "A"`

`I_2= 0.4293\ "A"`

`I_3= 0.0005\ "A"`

`I_4= -0.2224\ "A"`

`I_5= -0.2785\ "A"`

`I_6= 0.2112 \ "A"`

`I_7= 0.2091 \ "A"`

### Exercise 3

We want 10 L of gasoline containing 2% additive. We have drums of the following:

Gasoline without additive

Gasoline with 5% additive

Gasoline with 6% additive

We need to use 4 times as much pure gasoline as 5% additive gasoline. How much of each is needed?

Always check your solutions!

Answer

Let

x= no. of litres of pure gasoline

y= no. of litres of 5% gasoline

z= no. of litres of 6% gasoline

From the first sentence, we have:

`x + y + z = 10`

The second sentence gives us:

We get NO additive from the pure gasoline.

We get (5% of *y*) L of additive from the second drum.

We get (6% of *z*) L of additive from the third drum.

We NEED 2% of 10 L of additive = 0.2 L = 200 mL.

So

`0.05y + 0.06z = 0.2`

Multiplying through by 100 gives us:

`5y + 6z = 20`

The second last sentence gives us:

`x = 4y`

We can write this as:

`x - 4y = 0`

This gives us the set of simultaneous equations:

x+y+z= 105

y+ 6z= 20

x− 4y= 0

So

`A=((1,1,1),(0,5,6),(1,-4,0))`, `\ C=((10),(20),(0))`

Using Scientific Notebook for the inverse:

`((1,1,1),(0,5,6),(1,-4,0))^-1` `=((0.96,-0.16,0.04),(0.24,-0.04,-0.24),(-0.2,0.2,0.2))`

Multiplying the inverse by matrix *C*:

`((0.96,-0.16,0.04),(0.24,-0.04,-0.24),(-0.2,0.2,0.2))((10),(20),(0))` `=((6.4),(1.6),(2))`

So we have `6.4` L of pure gasoline, `1.6` L of 5% additive and `2` L of 6% additive.

Is it correct?

`6.4 + 1.6 + 2 = 10` L [Checks OK]

`5% xx 1.6 + 6% xx 2 = 200` mL [Checks OK]

`4 × 1.6 = 6.4` [Checks OK]

### Exercise 4

This statics problem was presented earlier in Section 3: Matrices.

From the diagram, we obtain the following equations (these equations come from statics theory):

**Vertical forces: **

F_{1} sin 69.3° − F_{2} sin 71.1° − F_{3} sin 56.6° + 926 = 0

**Horizontal forces: **

F_{1} cos 69.3° − F_{2} cos 71.1° + F_{3} cos 56.6° = 0

**Moments:**

7.80 F_{1} sin 69.3° − 1.50 F_{2} sin 71.1° − 5.20 F_{3} sin
56.6° = 0

Using matrices, find the forces F_{1}, F_{2} and F_{3}.

Answer

We write the first equation so that the constant term is on the right hand side:

F

_{1}sin 69.3° − F_{2}sin 71.1° − F_{3}sin 56.6° = −926

In matrix form, we write the equations as:

`((sin 69.3°,-sin 71.1°,-sin 56.6°),(cos 69.3°,-cos 71.1°,cos 56.6°),(7.80 sin 69.3°,-1.50 sin 71.1°,-5.20 sin 56.6°))((F_1),(F_2),(F_3))`

`=((-926),(0),(0))`

So the solution for the system is:

`((F_1),(F_2),(F_3))=((sin 69.3°,-sin 71.1°,-sin 56.6°),(cos 69.3°,-cos 71.1°,cos 56.6°),(7.80 sin 69.3°,-1.50 sin 71.1°,-5.20 sin 56.6°))^-1((-926),(0),(0))`

`=((425.5),(1079.9),(362.2))`

So

`F_1= 425.5\ "N"`

`F_2= 1079.9\ "N"`

`F_3= 362.2\ "N"`

This is very easy and quick in Scientific Notebook, Matlab or any other computer algebra system!