This method can be applied even if the coefficient matrix is singular matrix and rectangular matrix.

**Gaussian Elimination Method**

This method can be applied even if the coefficient matrix is
singular matrix and rectangular matrix. It is essentially the method of
substitution which we have already seen. In this method, we transform the
augmented matrix of the system of linear equations into row-echelon form and
then by back-substitution, we get the solution.

Solve the following system of linear equations, by Gaussian
elimination method :

4*x *+
3*y *+ 6*z *= 25, *x *+ 5 *y *+ 7*z *= 13, 2*x *+ 9 *y *+ *z *= 1.

Transforming the augmented matrix to echelon form, we get

The equivalent system is written by using the echelon form:

x + 5y + 7z = 13 , â€¦ (1)

17y + 22z = 27 , â€¦ (2)

199z = 398 . â€¦ (3)

Substituting
z = 2, y = -1 in (1), we get x = 13 -
5 Ã— (âˆ’1 ) âˆ’ 7 Ã— 2 = 4 .

So,
the solution is ( x =4, y = - 1, z = 2 ).

**Note. **The above method of going from the
last equation to the first equation is called the **method of back substitution**.

The upward speed
*v*(*t*) of a rocket at time t is
approximated by v(t) = at^{2} + bt + c, 0 â‰¤ t â‰¤ 100 where a, b, and c are constants. It
has been found that the speed at times t = 3, t = 6 , and t = 9 seconds are
respectively, 64, 133, and 208 miles per second respectively. Find the speed at
time t = 15 seconds. (Use Gaussian elimination method.)

Since *v*(3) =64, *v*(6) = 133 and *v*(9) = 208 , we get the following system of linear equations

9a +3b + c = 64 ,

36a + 6b + c = 133,

81a + 9b + c = 208 .

We solve the above system of linear equations by Gaussian
elimination method.

Reducing the augmented matrix to an equivalent row-echelon form by
using elementary row operations, we get

Writing the equivalent equations from the row-echelon matrix, we
get

9a + 3b + c = 64, 2b + c = 41,
c= 1.

By back substitution, we get

So, we get v (t) = 1/3 t^{2}_{ } + 20t + 1.

Hence, *v*(15) = 1/3 (225)
+ 20(15) + 1 = 75 + 300 + 1 = 376

Tags : Definition, Solved Example Problems , 12th Mathematics : UNIT 1 : Applications of Matrices and Determinants

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12th Mathematics : UNIT 1 : Applications of Matrices and Determinants : Matrix: Gaussian Elimination Method | Definition, Solved Example Problems

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