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WEEK 8

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WEEK 8

SYSTEMS OFEQUATIONS

DETERMINANTS AND CRAMER’S RULE

- At the end of this session , you will be able to:
- Evaluate a second order determinant.
- Evaluate a third-order determinant.
- Evaluate higher order determinants.
- Solve a system of linear equations in two variables using Cramer’s rule.
- Solve a system of linear equations in three variables using Cramer’s rule.
- Use determinant’s to identify inconsistent systems and systems with dependent equations.

- Determinants
- Solving a system of linear equations using determinants.
2.1 Solution of system of linear equations in two variables by Cramer’s rule

2.2 Solution of System of Linear Equationsin three variables by Cramer’s rule

3. Summary

- A determinant is a real number associated with every square matrix.
- Corresponding to each square matrix
there is associated an expression, called the determinant of A, denoted by det A or |A|, written as

- A matrix is an arrangement of numbers and so it has no fixed value, while each determinant has a fixed value.
- A determinant having n rows and n columns is known as a determinant of order n.
- NOTE: 1. Only square matrices have determinants. The determinants of non square
matrices are not defined.

2.Determinants are useful in solving a system of linear equations and help in determining if the system is consistent or inconsistent.

- Determinant of square matrix of order 1: If A = [a11] is a square matrix of order 1, then the determinant of A is defined as |A| = a11

- Determinant of square matrix of order 2: If is a square matrix of order 2, then
the expression a11a22 – a12a21 is defined as the determinant of A, that is,

- We also say that the value of second order determinant
- Thus, the determinant of a square matrix of order 2 is equal to the product of the diagonal elements minus the product of off-diagonal elements.
- Example: Let us evaluate
From the above definition, we have

- NOTE:
- The determinant is a real number, it is not a matrix.
- The determinant can be a negative number.
- It is not associated with absolute value at all except that they both use vertical lines.
- The determinant only exists for square matrices (2×2, 3×3, ... n×n). The determinant of a 1×1 matrix is that single value in the determinant.

- Determinant of order 3 or more:
For finding the value of a determinant of order 3 or more, we need the following definitions:

- Minor : The minor of an element aijin |A| is defined as the value of the determinant obtained by deleting the ith row and jth column of |A|, and is denoted by Mij.
- Cofactor: The cofactor Cijof an element is defined as Cij = (-1)i+j Mij.
- Now let us find the minors and cofactors of the elements of the determinant
Let Mijdenote the minor of aijin |A|.

Now a11occurs in the first row and first column. In order to find the minor of a11,we delete the first row and first column of |A|.

The minor M11 of a11 is given by,

Next to find the minor of a12, that is, the element in the first row, second column, we delete the first row and second column of |A|.

The minor M12 of a12is given by,

Next let us find M12 , that is, we have to find the minor of a12

So, M12 = -3

Similarly, M21 = Minor of a21 = -7

We have to find the minor of a11, so we delete the first row and second column

We have to find the minor of a12, so we delete the first row and second column

- Similarly, we have
- Similarly, we may obtain the minor of each of the remaining elements.
- Now, if we denote the cofactor of aij by Cij , then
C11 is the cofactor of a11, that is, cofactor of the element in the first row, first column.

By definition of cofactor, Cij = (-1)i+j Mij.

C11 = (-1)1+1M11

= (-1)2 M11

= M11

= (Substituting the value of M11)

Similarly,

Similarly, the cofactor of each of the remaining elements of |A| can be determined.

- Example: Let us find the minor and cofactor of each element of
- The minors of the elements of |A|are given by

Now let us find the cofactors of the corresponding elements of |A|.

From the definition of cofactors, Cij = (-1)i+j Mij.

- Value of a Determinant:The value of a determinant is the sum of the products of the elements of a row (or a column) with their corresponding cofactors.
- We find the determinant of a matrix of order three or more by expanding along any arbitrarily chosen row or column.
- Expansion of a Determinant: Expanding the determinant of order three along first row, we have
= a11. (Its cofactor) + a12. (Its cofactor) + a13. (Its cofactor)

= a11. C11 + a12. C12 + a13. C13

=a11. M11 - a12. M12 + a13. M13 (As C12 = -M12)

=

=

We may expand by any row or column.

NOTE: 1. We can expand a determinant by minors about any row or column. We use alternating plus and minus signs to precede the numerical factors of the minors, keeping in view the following sign array:

2. If a row or column of a determinant consists of all zeros, the value of the determinant is zero.

- Now let us understand how to evaluate a determinant of matrix of order three with help of an example:
- Let us evaluate the determinant of
- Expanding the given determinant about first row, we get
= 3.{(1).(-1) - (-5).(5)} – 0{(2).(1) – (-5).(2)} + 0.(2. 5 + 1. 2)

= 3.{-1 + 25} – 0 + 0

= 3 . 24 = 72

- Expanding the given determinant about first row, we get

- NOTE: To expand a determinant choose the row or column with the most zeros in it.Since each minor or cofactor is multiplied by the element in the matrix, picking a row or column with lots of zeros in it means that you will be multiplying by a lot of zeros. In fact, if the element is zero, you don't need to even find the minor or cofactor.
- Let us evaluate the determinant of 4X4 matrix
- To evaluate the determinant of a square matrix of order 4 or more we follow the same procedure as discussed in evaluating the determinant of a square matrix of order 3.
= 1.(its cofactor) + 2.(its cofactor) + (-1).(its cofactor)

+ 3 .(its cofactor)

- To evaluate the determinant of a square matrix of order 4 or more we follow the same procedure as discussed in evaluating the determinant of a square matrix of order 3.

Solving further, we get,

= 1 . C11 + 2 . C12 + (-1) . C13 + 3 . C14

= 1 . M11 + 2 .(-M12) + (-1) . M13 + 3 . (-M)14 (Cij = (-1)i+j Mij).

= 1 . M11 - 2 .M12 + (-1) . M13 - 3 . M14

Now evaluating each of the above determinants of order 3 as explained before, we get the following result:

|A|= 1.(16) – 2.(12) + (-1).(-11) – 3.(14)

= 16 – 24 + 11 – 42

= -39

- 2.1 SOLUTION OF SYSTEM OF LINEAR EQUATIONS IN TWO VARIABLES BY CRAMER’S RULE:
We now intend to solve a system of simultaneous linear equations by Cramer’s rule named after the Swiss mathematician Gabriel Cramer.

Cramer’s Rule: The solution of the system of simultaneous linear equations

a1x + b1y = c1

a2x + b2y = c2

is given by

where provided that D 0.

REMARK: Here, is the determinant of the coefficient matrix

NOTE: 1. Three different determinants are used to find x and y. The determinants in the denominators for x and y are identical. The determinants in the numerators for x and y differ.

Dx and Dy are obtained by replacing the x-coefficients and y-coefficients in D respectively with the constants c1 and c2.

2. Dx, the determinant in the numerator of x, is obtained by replacing the x-coefficients in D, a1 and a2, with the constants on the right side of the equations, c1 and c2.

3. Dy, the determinant in the numerator of y, is obtained by replacing the y-coefficients in D, b1 and b2, with the constants on the right side of the equations, c1 and c2.

- Conditions for Consistency:
- For a system of two simultaneous linear equations with two unknowns we have the following conditions:
- If D 0 then the system is consistent and has a unique solution given by
- If D = 0, Dx = 0, and Dy = 0, the system is consistent and has infinite number of solutions. The equations in the system are dependent.
- If D = 0, and one of Dx and Dy is non-zero, then the system is inconsistent.

- For a system of two simultaneous linear equations with two unknowns we have the following conditions:

- Example: Solve the following system of equations with the help of determinants
2x – y = 17

3x + 5y = 6

We know , substituting the values from the given system of equation, we get,

Now, , substituting the values from the given system of equation, we get,

Next,

Substituting the values from the given system of equations, we have,

Now we have found the values of all the three determinants, using Cramer’s rule, we have,

Substituting the values, we get,

Hence, the solution set for the given system of equations is {(7, -3)}.

CHECK: We can always check the solution (7, -3) by substituting these values into the original system of equations.

- Additional Examples:

- 2.2 SOLUTION OF SYSTEM OF LINEAR EQUATIONS IN THREE VARIABLES BY CRAMER’S RULE:
The solution of the system of linear equations

a1x + b1y + c1z = d1

a2x + b2y + c2z = d2

a3x + b3y + c3z = d3

is given by

where

provided that D 0

REMARK: Here D is the determinant of the coefficient matrix.

- Conditions for Consistency: For a system of three simultaneous linear equations in three unknowns,
- If D 0, then the given system of equations is consistent and has a unique solution given by
- If D = 0 and Dx = Dy = Dz = 0, then the system of equations is consistent with infinitely many solutions.
- If D = 0 and at least one of the determinants Dx , Dy , Dz is non-zero, then the given system of equations is inconsistent.

- Example: Let us solve the following system of equations using Cramer’s rule:
5x – 7y +z = 11

6x – 8y – z = 15

3x + 2y – 6z = 7

We know

(Substituting the values from the given system of equations)

Expanding about first row, we get,

Similarly,

So, by Cramer’s rule, we have,

Hence, the solution set for the given system of equations is {(1, -1, -1)}.

The solution (1, -1, -1) can be checked by substituting the values into the original system of equations.

- Additional Examples:

Let us recall what we have learnt so far:

- A determinant is a real number associated with every square matrix.
- The value of second order determinant
- Minor : The minor of an element aijin |A| is defined as the value of the determinant obtained by deleting the ith row and jth column of |A|, and is denoted by Mij.
- Cofactor: The cofactor Cijof an element id defined as Cij = (-1)i+j Mij.
- Value of the third order determinant
=

- Cramer’s Rule: The solution of the system of simultaneous linear equations
a1x + b1y = c1

a2x + b2y = c2 is given by

- Cramer’s Rule: The solution of the system of linear equations
a1x + b1y + c1z = d1

a2x + b2y + c2z = d2

a3x + b3y + c3z = d3 is given by

- Conditions for Consistency:
- For a system of two simultaneous linear equations with two unknowns we have the following conditions:
- If D 0 then the system is consistent and has a unique solution given by
- If D = 0, Dx = 0, and Dy = 0, the system is consistent and has infinite number of solutions.
- If D = 0, and one of Dx and Dy is non-zero, then the system is inconsistent.

- For a system of three simultaneous linear equations in three unknowns,
- If D 0, then the given system of equations is consistent and has a unique solution given by
- If D = 0 and Dx = Dy = Dz = 0, then the system of equations is consistent with infinitely many solutions.
- If D = 0 and at least one of the determinants Dx , Dy , Dz is non-zero, then the given system of equations is inconsistent.

- For a system of two simultaneous linear equations with two unknowns we have the following conditions: