Matrix And Determinant - Question 2(On System Of Linear Equation)

Introduction

Topic: Matrix and Determinant
Exam: 12th Boards Mathematics
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What is a Matrix?

A matrix is a rectangular array (or table) of numbers or symbols, arranged in rows and columns.
It is denoted by a capital letter.
Example: A =

Types of Matrices

Zero Matrix:
- All elements are 0.
- Example:
Square Matrix:
- Number of rows equal to number of columns.
- Example:
Row Matrix:
- Single row and multiple columns.
- Example:
Column Matrix:
- Single column and multiple rows.
- Example:
Diagonal Matrix:
- All non-diagonal elements are zeros.
- Example:

Types of Matrices (contd.)

Identity Matrix:
- Diagonal elements are 1 and non-diagonal elements are 0.
- Example:
Upper Triangular Matrix:
- All elements below diagonal are zeros.
- Example:
Lower Triangular Matrix:
- All elements above diagonal are zeros.
- Example:
Symmetric Matrix:
- A matrix is equal to its transpose.
- Example:
Skew-Symmetric Matrix:
- A matrix is equal to the negative of its transpose.
- Example:

Addition and Subtraction of Matrices

Matrices can be added or subtracted if they have the same dimensions.
To add or subtract matrices, add or subtract corresponding elements.
Example:

A =

B =

A + B =
Similarly, matrices can be subtracted using the same rule.

Scalar Multiplication of a Matrix

Scalar multiplication involves multiplying each element of a matrix by a constant (scalar).
To multiply a matrix by a scalar, multiply each element by the scalar value.
Example: A =

c = 3

c * A =

Matrix Multiplication

Matrix multiplication is different from scalar multiplication.
In matrix multiplication, the product of two matrices is calculated by taking the dot product of rows and columns.
Example: A =

B =

A * B =
Note that for matrix multiplication, the number of columns in the first matrix should be equal to the number of rows in the second matrix.

Properties of Matrix Multiplication

Matrix multiplication is associative: (AB)C = A(BC)
Matrix multiplication is not commutative: AB ≠ BA
Multiplying a matrix by the identity matrix gives the same matrix: AI = A and IA = A
Matrix multiplication is distributive over addition: A(B + C) = AB + AC

Properties of Matrix Multiplication (contd.)

Scalar multiplication can be distributed to matrix multiplication: (cA)B = c(AB) = (AB)c
Transpose of a product: (AB)^T = B^T * A^T

System of Linear Equations: Introduction

A system of linear equations is a set of equations in which all the equations are linear (i.e., the highest power of the variable is 1).
The solution to a system of linear equations is a set of values that satisfy all the equations simultaneously.
A system of linear equations can have three possibilities for the solution:
- A unique solution
- Infinitely many solutions
- No solution

Solving a System of Linear Equations: Methods

There are three main methods for solving a system of linear equations:
1. Substitution method
  - Solve one equation for one variable and substitute it into the other equation.
  - Continue substituting until one variable is eliminated, and then solve for the remaining variable.
  - Example:
    - Equation 1: 2x + y = 5
    - Equation 2: 3x - 2y = 8
    - Solve Equation 1 for x: x = (5 - y)/2
    - Substitute x in Equation 2: 3((5 - y)/2) - 2y = 8
    - Simplify and solve for y, then substitute y to find x.
2. Elimination method
  - Multiply one or both equations by a factor to make the coefficients of one variable opposite in sign.
  - Add or subtract the equations to eliminate one variable.
  - Solve for the remaining variable, then substitute back to find the other variable.
  - Example:
    - Equation 1: 2x + y = 5
    - Equation 2: 3x - 2y = 8
    - Multiply Equation 1 by 2: 4x + 2y = 10
    - Add Equation 2 and the new Equation 1: 7x = 18
    - Solve for x, then substitute x to find y.

Solving a System of Linear Equations: Methods (contd.)

There are three main methods for solving a system of linear equations (continued): 3. Matrix method (using augmented matrices) - Write the system of equations as an augmented matrix. - Perform row operations to reduce the augmented matrix to row-echelon form, then to reduced row-echelon form. - The resulting matrix will represent the solution to the system of equations. - Example: - Equation 1: 2x + y = 5 - Equation 2: 3x - 2y = 8 - Augmented matrix: [2 1 | 5; 3 -2 | 8] - Perform row operations to reduce to reduced row-echelon form: [1 0 | 2; 0 1 | 1] - Solution: x = 2, y = 1

Solving a System of Linear Equations: Practice Problem

Solve the following system of equations using the method of your choice:
- Equation 1: 3x - 2y = 5
- Equation 2: 2x + 4y = -3
- Solution:
  - Substitution method:
    - Solve Equation 1 for x: x = (5 + 2y)/3
    - Substitute x in Equation 2: 2((5 + 2y)/3) + 4y = -3
    - Simplify and solve for y, then substitute y to find x.
  - Elimination method:
    - Multiply Equation 1 by 2 and Equation 2 by 3: 6x - 4y = 10, 6x + 12y = -9
    - Subtract the two equations to eliminate x, then solve for y.
    - Substitute y to find x.
  - Matrix method:
    - Write the augmented matrix: [3 -2 | 5; 2 4 | -3]
    - Perform row operations to reduce to reduced row-echelon form, then solve for x and y.

Determinants: Definition and Properties

The determinant is a scalar value that can be computed from the elements of a square matrix.
Properties of determinants:
1. The determinant of the identity matrix is 1.
2. If a matrix has two identical rows (or columns), its determinant is 0.
3. If a row or column is multiplied by a scalar, the determinant is multiplied by the same scalar.
4. If two rows or columns are interchanged, the determinant changes sign.
5. If rows (or columns) are added or subtracted, the determinant remains the same.
6. The determinant of a diagonal matrix is the product of its diagonal elements.

Determinants: 2x2 Matrix

For a 2x2 matrix, A = [a b; c d], the determinant is given by the formula:
- det(A) = ad - bc
Example:
- A = [2 1; 3 4]
- det(A) = (2)(4) - (1)(3) = 8 - 3 = 5

Determinants: 3x3 Matrix

For a 3x3 matrix, A = [a b c; d e f; g h i], the determinant is given by the formula:
- det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
Example:
- A = [2 1 3; 0 -1 2; 1 0 -3]
- det(A) = (2)(-1)(-3) - (1)(2)(1) + (3)(0)(0) = 6 + 2 + 0 = 8

Determinants: Cramer’s Rule

Cramer’s Rule is a method for solving a system of linear equations using determinants.
For a system of linear equations:
- Equation 1: a1x + b1y = c1
- Equation 2: a2x + b2y = c2
The solution can be found using the formulas:
- x = det(Ax) / det(A)
- y = det(Ay) / det(A)
Where A is the coefficient matrix, Ax is obtained by replacing the x-column of A with the constant terms, and Ay is obtained by replacing the y-column of A with the constant terms.

Determinants: Cramer’s Rule Example

Solve the following system of equations using Cramer’s Rule:
- Equation 1: 2x - 3y = 7
- Equation 2: 4x + 5y = 9
Solution:
- A = [2 -3; 4 5], Ax = [7 -3; 9 5], Ay = [2 7; 4 9]
- Calculate det(A), det(Ax), and det(Ay) using the determinant formula.
- Substitute into Cramer’s Rule: x = det(Ax) / det(A), y = det(Ay) / det(A)
- Simplify to find the values of x and y.

Conclusion

In this lecture:
- We learned about system of linear equations and the different methods for solving them.
- We explored determinants and their properties, as well as formulas for calculating determinants of 2x2 and 3x3 matrices.
- We discussed Cramer’s Rule as a method for solving systems of linear equations using determinants.
Practice solving additional problems to strengthen your understanding of these topics.
Keep practicing and revise regularly to excel in your exams!

System of Linear Equations: Practice Problem 2

Solve the following system of equations using the method of your choice:
- Equation 1: x + 2y = 3
- Equation 2: 2x + 3y = 5
- Solution:
  - Substitution method:
    - Solve Equation 1 for x: x = 3 - 2y
    - Substitute x in Equation 2: 2(3 - 2y) + 3y = 5
    - Simplify and solve for y, then substitute y to find x.
  - Elimination method:
    - Multiply Equation 1 by 2 and Equation 2 by -1: 2x + 4y = 6, -2x - 3y = -5
    - Add the two equations to eliminate x, then solve for y.
    - Substitute y to find x.
  - Matrix method:
    - Write the augmented matrix: [1 2 | 3; 2 3 | 5]
    - Perform row operations to reduce to reduced row-echelon form, then solve for x and y.

Determinants: 4x4 Matrix

For a 4x4 matrix, A = [a b c d; e f g h; i j k l; m n o p], the determinant is given by the formula:
- det(A) = a(fkp - gjo - eln - hmj) - b(ekp - gbo - dln - hkn) + c(ejo - fbo - dmj - hlo) - d(ejn - fkn - dgm - hlm)
Example:
- A = [1 2 3 4; 5 6 7 8; 9 10 11 12; 13 14 15 16]
- det(A) = (1)(-72) - (2)(-56) + (3)(-40) - (4)(-24) = -72 + 112 - 120 + 96 = 16

Determinants: Properties and Applications

More properties of determinants: 7. If a matrix is multiplied by its inverse, the determinant is 1. 8. If a matrix is singular (its determinant is 0), it is not invertible. 9. If a matrix has a row (or column) of zeros, its determinant is 0. 10. The determinant of a product of matrices equals the product of their determinants.
Determinants have various applications in mathematics, including solving systems of linear equations, finding the area or volume of geometric shapes, and determining whether a matrix is invertible.

Matrix Inverse

The inverse of a square matrix A, denoted as A^(-1), is the matrix such that A * A^(-1) = I, where I is the identity matrix.
Not all matrices have inverses. A matrix A is invertible if and only if its determinant is non-zero.
Example:
- A = [2 -1; 3 4]
- Calculate the determinant of A.
- If the determinant is non-zero, find the adjoint of A.
- Divide the adjoint by the determinant to find the inverse of A.

Matrix Inverse Example

Find the inverse of the matrix A = [4 3; 2 1].
Solution:
- Calculate the determinant of A.
- If the determinant is non-zero, find the adjoint of A.
- Divide the adjoint by the determinant to find the inverse of A.
- Check if the product of A and its inverse equals the identity matrix.

Matrix Inverse: Practice Problem

Find the inverse of the matrix A = [1 2; 3 4].
Solution:
- Calculate the determinant of A.
- If the determinant is non-zero, find the adjoint of A.
- Divide the adjoint by the determinant to find the inverse of A.
- Check if the product of A and its inverse equals the identity matrix.

Determinants and Cramer’s Rule

Cramer’s Rule can be used to solve systems of linear equations using determinants.
If the determinant of the coefficient matrix is non-zero, the system has a unique solution.
Example:
- Equation 1: 2x + 3y = 7
- Equation 2: 4x - y = -3
- Set up the coefficient matrix A and the constant matrix C.
- Calculate the determinant of A. If non-zero, calculate the determinants of Ax and Ay.
- Substitute the determinants into Cramer’s Rule equations to find x and y.

Determinants and Cramer’s Rule: Practice Problem

Use Cramer’s Rule to solve the system of equations:
- Equation 1: 3x + 2y = 8
- Equation 2: 5x - 4y = 3
Solution:
- Set up the coefficient matrix A and the constant matrix C.
- Calculate the determinant of A. If non-zero, calculate the determinants of Ax and Ay.
- Substitute the determinants into Cramer’s Rule equations to find x and y.

Determinants and Area of a Triangle

Determinants can be used to find the area of a triangle given its vertices.
Given three points A(x1, y1), B(x2, y2), and C(x3, y3), the area of the triangle ABC can be calculated using the formula:
- Area = 0.5 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
Example:
- A(2, 1), B(5, -3), C(7, 4)
- Calculate the area of triangle ABC using the determinant formula.

Summary and Final Remarks

In this lecture, we covered the concept of matrix inverse, determinants, and their properties.
We learned about Cramer’s Rule for solving systems of linear equations.
Determinants have various applications in mathematics, including finding areas and volumes.
Matrix operations and determinants are important in 12th Boards Mathematics, so practice solving problems regularly.
Good luck with your exam preparations! Keep up the hard work and stay confident.