This rule can be applied only when the coefficient matrix is a square matrix and non-singular.

**Cramerâ€™s Rule**

This rule can be applied only when the coefficient matrix is a
square matrix and non-singular. It is explained by considering the following
system of equations:

where the coefficient matrix is non-singular. Then

Let us put Î” = . Then, we have

**Note**

Replacing the first column elements *a*_{11} , *a*_{21}
, *a*_{31} of Î” with *b*_{1}
, *b*_{2} , *b*_{3} respectively, we get Î”_{1}. Replacing the second column elements *a*_{12} , *a*_{22}
, *a*_{32} of Î” with *b*_{1}
, *b*_{2} , *b*_{3} respectively, we get Î”_{2} . Replacing the third column elements *a*_{13} , *a*_{23}
, *a*_{33} of Î” with *b*_{1}
, *b*_{2} , *b*_{3} respectively, we get Î”_{3}.

If Î” = 0, Cramerâ€™s rule
cannot be applied.

Solve, by Cramerâ€™s rule, the system of equations

*x*_{1} âˆ’ *x*_{2} = 3, 2*x*_{1}
+ 3*x*_{2} + 4*x*_{3} = 17, *x*_{2}
+ 2*x*_{3} = 7.

First we evaluate the determinants

So, the solution is (*x _{1}* =
2,

In a T20 match, Chennai Super Kings needed just 6 runs to win with
1 ball left to go in the last over. The last ball was bowled and the batsman at
the crease hit it high up. The ball traversed along a path in a vertical plane
and the equation of the path is *y *=
*ax*^{2} +
*bx *+ *c *with respect
to a *xy *-coordinate system in the vertical plane
and the ball traversed through
the points (10,8),
(20,16), (30,18) , can you conclude that Chennai Super Kings won the match?

Justify your answer. (All distances are measured in metres and
the meeting point of the plane of the path with the farthest boundary line is (70,
0).)

The
path *y
*= *ax*2
+ *bx *+ *c *passes through the points
(10,8), (20,16), (40, 22) . So, we get the system of equations 100*a *+
10*b** *+ *c *=
8, 400*a *+
20*b *+ *c *= 16,1600*a *+ 40*b *+
*c
*= 22. To apply
Cramerâ€™s rule, we find

When *x *= 70, we get *y *= 6. So,
the ball went by 6 metres high over the boundary line and it is impossible for a fielder
standing even just before the boundary line to jump and catch the ball. Hence
the ball went for a super six and the Chennai Super Kings won the match.

Tags : Definition, Formulas, 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 : Matrices: Cramerâ€™s Rule | Definition, Formulas, Solved Example Problems

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