Matrix And Determinant

Introduction to Matrix and Determinant
Definition of Matrix and Determinant
Types of Matrices
- Zero Matrix
- Square Matrix
- Diagonal Matrix
- Identity Matrix
Basic Operations on Matrices
- Addition
- Subtraction
- Scalar Multiplication
Properties of Matrices

Matrix and Determinant

Types of Matrices (contd.)
- Transpose of a Matrix
- Symmetric Matrix
- Skew-Symmetric Matrix
Determinant
- Definition of Determinant
- Properties of Determinants
Calculating Determinants
- 2x2 Matrix
- 3x3 Matrix
Inverse of a Matrix
- Definition of Inverse
- Finding Inverse using Determinants
Applications of Matrices and Determinants
- Solving Systems of Linear Equations
- Transformations in Coordinate Geometry

Matrix and Determinant

Introduction to Matrix and Determinant

Matrix: A rectangular array of numbers, symbols, or expressions arranged in rows and columns.
Determinant: A value associated with a square matrix that can be computed from its entries.
Matrices are widely used in various fields of mathematics, engineering, computer science, economics, and physics.
Determinants provide essential information about the properties and behavior of systems described by matrices.
Understanding matrices and determinants is crucial for solving problems involving linear equations, transformations, and more.

Definition of Matrix and Determinant

Matrix:
- Rows: Horizontal elements in a matrix.
- Columns: Vertical elements in a matrix.
- Size: Number of rows x Number of columns.
Determinant:
- For a 2x2 matrix: ad - bc.
- For a 3x3 matrix: a(ei - fh) - b(di - fg) + c(dh - eg).

Matrix and Determinant

Types of Matrices

Zero Matrix

All elements of the matrix are zero.
Denoted by the symbol O.

Square Matrix

Number of rows is equal to the number of columns.
Denoted by the symbol A.

Diagonal Matrix

All non-diagonal elements are zero.
Diagonal elements may or may not be zero.
Example: A = [3, 0, 0; 0, 5, 0; 0, 0, 7].

Identity Matrix

A square matrix with diagonal elements all equal to 1 and other elements equal to 0.
Denoted by the symbol I.
Example: I = [1, 0, 0; 0, 1, 0; 0, 0, 1].

More types of matrices to be covered in the next slide.

Matrix and Determinant

Types of Matrices (contd.)

Transpose of a Matrix

The transpose of a matrix is obtained by interchanging its rows into columns and columns into rows.
Denoted by the symbol A^T.

Symmetric Matrix

If the transpose of a matrix is equal to the original matrix, it is called a symmetric matrix.
Example: A = [1, 2, 3; 2, 4, 5; 3, 5, 6].

Skew-Symmetric Matrix

If the transpose of a matrix is equal to the negative of the original matrix, it is called a skew-symmetric matrix.
Example: A = [0, -2, 3; 2, 0, -5; -3, 5, 0].

Determinant

Definition of Determinant

The determinant of a square matrix is a scalar value that can be calculated using specific rules and formulas.
It provides information about the matrix’s invertibility, properties, and behavior in various mathematical operations.

Matrix and Determinant

Determinant

Properties of Determinants

The determinant of a matrix and its transpose are equal: det(A) = det(A^T).

If a matrix has a row or column consisting of all zeros, then its determinant is zero.

The determinant of a matrix is zero if any two rows or columns are proportional.

If two rows or columns of a matrix are interchanged, the determinant changes its sign.

The determinant of a matrix multiplied by a scalar is equal to the determinant of the scalar multiplied by the matrix.

The determinant of the product of two matrices is equal to the product of their determinants.

If a matrix has two identical rows or columns, its determinant is zero.

Calculating Determinants

Determinant of a 2x2 matrix: ad - bc.
Determinant of a 3x3 matrix: a(ei - fh) - b(di - fg) + c(dh - eg).

Matrix and Determinant

Inverse of a Matrix

Definition of Inverse

The inverse of a square matrix A is denoted by A^(-1).
If A^(-1) exists, it satisfies the property: A x A^(-1) = I, where I is the identity matrix.
A non-square matrix does not have an inverse.

Finding Inverse using Determinants

Calculate the determinant of the given matrix.

If the determinant is zero, the inverse does not exist.

If the determinant is non-zero, find the adjugate matrix.

Multiply the adjugate matrix by (1/determinant) to obtain the inverse of the matrix.

Applications of Matrices and Determinants

Solving Systems of Linear Equations
Transformations in Coordinate Geometry

Matrix and Determinant

Properties of Matrices

Addition:
- The sum of two matrices of the same size is obtained by adding the corresponding elements.
- Example: A + B = [a + d, b + e, c + f; g + j, h + k, i + l].
Subtraction:
- The difference between two matrices of the same size is obtained by subtracting the corresponding elements.
- Example: A - B = [a - d, b - e, c - f; g - j, h - k, i - l].
Scalar Multiplication:
- Multiplying a matrix by a scalar involves multiplying each element of the matrix by the scalar.
- Example: cA = [ca, cb, cc; cg, ch, ci], where c is a scalar.
Properties:
- Commutative Property: A + B = B + A.
- Associative Property: (A + B) + C = A + (B + C).
- Distributive Property: c(A + B) = cA + cB.

Matrix and Determinant

Properties of Matrices (contd.)

Zero Matrix:
- A matrix where all the elements are zero.
- Example: 0 = [0, 0, 0; 0, 0, 0].
Square Matrix:
- A matrix with an equal number of rows and columns.
- Example: A = [a, b, c; d, e, f; g, h, i].
Diagonal Matrix:
- A square matrix where all the non-diagonal elements are zero.
- Example: A = [a, 0, 0; 0, e, 0; 0, 0, i].
Identity Matrix:
- A square matrix with diagonal elements as 1 and other elements as 0.
- Example: I = [1, 0, 0; 0, 1, 0; 0, 0, 1].
Transpose of a Matrix:
- Obtained by interchanging rows and columns.
- Example: A^T = [a, d, g; b, e, h; c, f, i].
Symmetric Matrix:
- When the transpose of a matrix is equal to the original matrix.
- Example: A = [a, b, c; b, d, e; c, e, f].

Matrix and Determinant

Properties of Matrices (contd.)

Skew-Symmetric Matrix:
- When the transpose of a matrix is equal to the negative of the original matrix.
- Example: A = [0, -b, -c; b, 0, -e; c, e, 0].

Determinant

Calculating Determinants

Determinant of a 2x2 matrix:
- Given matrix A = [a, b; c, d].
- Determinant = ad - bc.
- Example: A = [2, 3; 4, 5]. Determinant = (2 * 5) - (3 * 4) = 10 - 12 = -2.
Determinant of a 3x3 matrix:
- Given matrix A = [a, b, c; d, e, f; g, h, i].
- Determinant = a(ei - fh) - b(di - fg) + c(dh - eg).
- Example: A = [1, 2, 3; 4, 5, 6; 7, 8, 9]. Determinant = (1 * (5 * 9 - 6 * 8)) - (2 * (4 * 9 - 6 * 7)) + (3 * (4 * 8 - 5 * 7)) = (1 * 3) - (2 * 9) + (3 * 6) = 3 - 18 + 18 = 3.

Matrix and Determinant

Inverse of a Matrix

Finding Inverse using Determinants

Given a square matrix A, to find its inverse A^(-1):

Calculate the determinant of A.

If the determinant is non-zero, A^(-1) exists.

Find the adjugate matrix of A by swapping the diagonal elements and changing the signs of all the other elements.

Multiply the adjugate matrix by (1/determinant) to get A^(-1).

If the determinant is zero, A^(-1) does not exist.

Example:

A = [2, 1; 3, 4]
Determinant of A = (2 * 4) - (1 * 3) = 8 - 3 = 5
Adjugate of A = [4, -1; -3, 2]
Inverse of A = (1/5) * [4, -1; -3, 2] = [4/5, -1/5; -3/5, 2/5]

Matrix and Determinant

Applications of Matrices and Determinants

Solving Systems of Linear Equations

Matrices can be used to solve systems of linear equations.
Given a system of equations:
- a₁x + b₁y + c₁z = d₁
- a₂x + b₂y + c₂z = d₂
- a₃x + b₃y + c₃z = d₃
This can be represented in matrix form: AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constants matrix.
By finding the inverse of A and multiplying it with B, we can solve for X: X = A^(-1)B.

Example:

To solve the system of equations:
- 2x + y + z = 6
- x - y + z = 2
- x + 3y - 2z = 4
Coefficient matrix A = [2, 1, 1; 1, -1, 1; 1, 3, -2]
Constants matrix B = [6; 2; 4]
X = A^(-1)B = [2, 1, 1; 1, -1, 1; 1, 3, -2]^(-1) * [6; 2; 4]
Calculate the inverse of A and multiply it with B to find the values of x, y, and z.

Matrix and Determinant

Applications of Matrices and Determinants (contd.)

Transformations in Coordinate Geometry

Matrices and determinants are used to describe and perform transformations in coordinate geometry.
Translation:
- A matrix can be used to represent the translation of a point in the coordinate plane.
- Example: A = [1, 0, dx; 0, 1, dy; 0, 0, 1], where dx and dy are the translation distances in the x and y directions respectively.
Rotation:
- A matrix can represent the rotation of a point around the origin.
- Example: A = [cosθ, -sinθ; sinθ, cosθ], where θ is the angle of rotation.
Scaling:
- A matrix can represent the scaling of a point about the origin or any other point.
- Example: A = [sx, 0; 0, sy], where sx and sy are the scaling factors in the x and y directions respectively.
Reflection:
- A matrix can represent the reflection of a point across the x-axis, y-axis, or any line.
- Example: Reflection across x-axis: A = [1, 0; 0, -1].

Matrix and Determinant

Applications of Matrices and Determinants (contd.)

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are important concepts in linear algebra.
Eigenvalues:
- Eigenvalues are scalars associated with a matrix.
- They represent the values by which a matrix stretches or contracts vectors.
- Example: For a matrix A, the eigenvalues λ satisfy the equation Ax = λx, where x is a non-zero vector.
Eigenvectors:
- Eigenvectors are non-zero vectors associated with eigenvalues.
- They represent the directions where the matrix only stretches or contracts, without changing direction.
- Example: For a matrix A and an eigenvalue λ, the eigenvectors x satisfy the equation Ax = λx.

Cramer’s Rule

Cramer’s rule is a method to solve systems of linear equations using determinants.
Given a system of equations:
- a₁x + b₁y + c₁z = d₁
- a₂x + b₂y + c₂z = d₂
- a₃x + b₃y + c₃z = d₃
Cramer’s rule states that the solutions to the equations can be found using determinants:
- x = (Dx / D)
- y = (Dy / D)
- z = (Dz / D)
Where Dx, Dy, and Dz are the determinants obtained by replacing the respective column in the coefficient matrix with the constants matrix, and D is the determinant of the coefficient matrix.

Example:

To solve the system of equations:
- 2x + 3y - z = 1
- x - 2y + 4z = -2
- 5x + y - 3z = 3
Coefficient matrix A = [2, 3, -1; 1, -2, 4; 5, 1, -3]
Constants matrix B = [1; -2; 3]
Calculate the determinant D and the determinants Dx, Dy, Dz.
Use Cramer’s rule to find the values of x, y, and z.