Determinants

Determinant of Matrices up to Order Three

**Determinant Method of Solving Simultaneous Equations**

When we have a system of two linear equations with two variables, the equations are known as simultaneous equations. We have already studied how to solve simultaneous equations by graphical method. But this method has some limitations and it fails to give the correct solutions in some cases. To overcome these limitations, we use the **determinant method**.

Let us first understand what a determinant is.

For any four numbers, *a*, *b*, *c* and *d*, value *ad− bc* can be represented as .

Here,*a*,* b* are in the first row and *c*, *d* are in the second row. Also, *a*,* c* are in the first column and *b*,* d* the second.

This type of representation of numbers or variables is called **determinant**.

This determinant has two rows and two columns, so it is a determinant of order two.

The value of this determinant is *ad − bc*. It can be observed that the value of the determinant shows the following pattern of multiplication and subtraction.

This pattern is helpful in finding the value of any determinant of order two.

There are determinants of higher orders as well but in this lesson, we will study about determinants of order two only.

Now,let us use the determinant method for finding the solution of a system of linear equations. This method is known as **Cramer’s rule**.

**Cramer’s rule:**

If we have two simultaneous equations, *a*_{1}*x+ b*_{1}*y= c*_{1}and *a*_{2}*x+ b*_{2}*y= c*_{2},where *a*_{1},*a*_{2},*b*_{1},*b*_{2},*c*_{1}and*c*_{2}are real numbers, such that *a*_{1}*b*_{2}− *a*_{2}*b*_{1}≠0 and *x *and *y *are variables, then values of *x *and *y *can be obtained as follows:

*x*= and *y* =

Here,D =,D_{x} = and D_{y} = .

This can be derived as follows:

*a*_{1}*x+ b*_{1}*y = c*_{1}…(1)

*a*_{2}*x+ b*_{2}*y = c*_{2}…(2)

On multiplying equation (1) by *b*_{2} and equation (2) by*b*_{1}:

*a*_{1}*b*_{2}*x *+ *b*_{1}*b*_{2}*y* = *c*_{1}*b*_{2}…(3)

*a*_{2}*b*_{1}*x+ b*_{1}*b*_{2}*y* = *c*_{2}*b*_{1}…(4)

On subtracting equation (4) from equation (3):

(*a*_{1}*b*_{2}*x*+ *b*_{1}*b*_{2}*y*) *−*(*a*_{2}*b*_{1}*x + b*_{1}*b*_{2}*y*)= *c*_{1}*b*_{2 }− *c*_{2}*b*_{1}

⇒(*a*_{1}*b*_{2}*− a*_{2}*b*_{1})*x*= *c*_{1}*b*_{2}*− c*_{2}*b*_{1}

Similarly,

Let us denote by D, by D_{x} and by D_{y}.

From equations (5) and (6), .

Let us consider a case when D = *a*_{1}*b*_{2}− *a*_{2}*b*_{1} = 0.

Thus,.

We know that division by 0 is not defined thus, values of *x* and *y *cannot be obtained in this case.

So,value of *a*_{1}*b*_{2} − *a*_{2}*b*_{1}can never be 0.

Let us go through a few examples to understand the application of this method.

**Example1:**

Find the value of the following determinants.

**Solution:**

We know that the solution of is *ad − bc*.

Now,let us solve the given determinants.

**Example2:**

If the value of is10, then what is the value of *x* in the given determinant?

**Solution:**

We have:

**Example 3:**

Solve the given simultaneous equations using Cramer’s rule.

5*x*− 2*y* = 2; 7*x* + 2*y* = 4

**Solution:**

The given equations are 5*x* − 2*y* = 2 and 7*x* + 2*y* = 4.

On comparing the coefficients of the given equations with that of the general equations, *a*_{1}*x+ b*_{1}*y= c*_{1}and *a*_{2}*x+ b*_{2}*y= c*_{2},we get *a*_{1}= 5, *b*_{1}= −2,*c*_{1}= 2, *a*_{2}= 7, *b*_{2}= 2 and *c*_{2}= 4.

Now,let us find D, D_{x} and D_{y}.

D=

⇒D =

⇒D = (5 × 2) −[7 × (− 2)]

⇒D = 10 + 14

⇒D = 24

Also,D_{x} =

⇒D_{x}=

⇒D_{x}= (2 × 2) − [4 × (−2)]

⇒D_{x}= 4 + 8

⇒D_{x}= 12

And,D_{y} =

⇒D_{y}=

⇒D_{y}= (5 × 4) − (7 × 2)

⇒D_{y}= 20 − 14

⇒D_{y}= 6

Now,

Thus,values of *x *and *y* for the given simultaneous equations are and respectively.

**Example4:**

Solve the following simultaneous equations using Cramer’s rule.

3*x*+ 2*y* − 4 = 6; 7*x *− 2*y* + 5 = 7

**Solution:**

The given equations are:

3*x*+ 2*y* − 4 = 6 and 7*x *− 2*y* + 5 = 7

⇒3*x*+ 2*y*= 6 + 4 and 7*x *−2*y*= 7 − 5

⇒3*x*+ 2*y*= 10 and 7*x *−2*y*= 2

On comparing the coefâ€¦

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