6. Matrices and Linear Equations
by M. Bourne
We wish to solve the system of simultaneous linear equations using matrices:
If we let
,
and
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-1AX = A-1C.
However, we know that A-1A = I, the Identity matrix. So we obtain
IX = A-1C.
But IX = X, so the solution to the system of equations is given by:
X = A-1C
See the box at the top of Inverse of a Matrix for more explanation about why this works.
Note: We cannot use CA-1 because matrix multiplication is not commutative.
Example - solving a system using the Inverse Matrix
Solve the system using matrices.
Always check your solutions!
Now let's see how LiveMath can do this for us.
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 LiveMath (or similar) to find inverses larger than 2×2.
Example - 3×3 System of Equations
Solve the system using matrix methods.
Did I mention? It's a good idea to always check your solutions.
Now let's see how LiveMath can do this for us.
Example - Electronics application of 3×3 System of Equations
Find the electric currents shown by solving the matrix equation (obtained using Kirschoff's law) arising from this circuit:
Exercise 1
The following equations are found in a particular circuit. Find the currents using matrix methods.
Exercise 2
Recall this problem from before? Now we know how to solve it, using inverse matrices on a computer.
The circuit equations, using Kirschoff's Law:
-26 = 72I1 - 17I3 - 35I4
34 = 122I2 - 35I3 - 87I7
-13 = 149I3 - 17I1 - 35I2 - 28I5 - 35I6 - 34I7
5 = 105I4 - 35I1 - 43I5
-27 = 105I5 - 28I3 - 43I4 - 34I6
24 = 141I6 - 35I3 - 34I5 - 72I7
-4 = 233I7 - 87I2 - 34I3 - 72I6
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!
Exercise 4
This statics problem was presented earlier in Section 3: Matrices.
From the diagram, we obtain the following equations:
Vertical forces:
F1 sin 69.3° − F2 sin 71.1° − F3 sin 56.6° + 926 = 0
Horizontal forces:
F1 cos 69.3° − F2 cos 71.1° + F3 cos 56.6° = 0
Moments:
7.80 F1 sin 69.3° − 1.50 F2 sin 71.1° − 5.20 F3 sin 56.6° = 0
Using matrices, we can find the forces F1, F2, & F3.
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