Determinants - Definition of Singular Matrix

• A matrix is said to be singular if its determinant is zero.
• The determinant of a matrix is denoted by |A| or det(A).
• If the determinant of a matrix is zero, it implies that the matrix is singular.
• A matrix is said to be non-singular if its determinant is non-zero.
• A non-singular matrix is also called an invertible or non-singular matrix.
• For a 2x2 matrix A = |a b| |c d|, if ad - bc = 0, then A is a singular matrix.
• For a 3x3 matrix A = |a b c| |d e f| |g h i| , if aei + bfg + cdh - ceg - bdi - afh = 0, then A is a singular matrix.

Determinants - Evaluation of Determinant

• The determinant of a matrix can be evaluated using various methods.
• The most common methods are cofactor expansion and row reduction.
• Cofactor expansion method involves expanding the determinant along a row or column by using the formula det(A) = a1C1 + a2C2 + … + anCn, where ai is the element and Ci is its cofactor.
• Row reduction method involves simplifying the matrix by performing elementary row operations until the matrix is in a triangular or diagonal form.
• The determinant of a matrix is always a scalar quantity.

Determinants - Properties of Determinants

• Determinant of a matrix remains unaffected if its rows and corresponding columns are interchanged.
• The determinant of a matrix is zero if any two rows (or columns) are identical.
• If a matrix has one row (or column) consisting of all zeros, then its determinant is zero.
• If each element of a row (or column) of a matrix is multiplied by a scalar k, then the determinant of the new matrix is k times the determinant of the original matrix.
• If two rows (or columns) of a matrix are proportional, then its determinant is zero.
• If two rows (or columns) of a matrix are interchanged, then the determinant still remains the same in magnitude but changes its sign.
• If all the elements of a row (or column) of a matrix are multiplied by k, then the determinant of the new matrix is also multiplied by k.

Determinants - Solving Linear Equations using Determinants

• Determinants can be used to solve a system of linear equations.
• Consider a system of equations in the form: a1x + b1y + c1z = d1 a2x + b2y + c2z = d2 a3x + b3y + c3z = d3
• The system of equations can be represented in matrix form as AX = B, where A is the coefficient matrix, X is the column matrix of variables, and B is the column matrix of constants.
• The determinant of A, denoted as |A| or det(A), can be calculated.
• If the determinant of A is non-zero, then the system has a unique solution.
• If the determinant of A is zero, then the system either has infinitely many solutions or no solutions, depending on the consistency of the equations.

Matrices - Introduction

• A matrix is a rectangular arrangement of numbers or symbols in rows and columns.
• The numbers or symbols in a matrix are called elements or entries.
• The size of a matrix is given by the number of rows and columns it contains.
• A matrix with m rows and n columns is said to be an m x n matrix, denoted as mxn matrix.
• A matrix with an equal number of rows and columns is called a square matrix.
• The elements of a matrix can be denoted by lowercase or uppercase letters.

Matrices - Types of Matrices

• Row matrix: A matrix with a single row and multiple columns is called a row matrix.
• Column matrix: A matrix with a single column and multiple rows is called a column matrix.
• Zero matrix: A matrix in which all elements are zero is called a zero matrix or null matrix.
• Diagonal matrix: A square matrix in which all the non-diagonal elements are zero is called a diagonal matrix. The diagonal elements may or may not be zero.
• Identity matrix: A square matrix in which all the diagonal elements are 1 and all the non-diagonal elements are zero is called an identity matrix, denoted by I.
• Symmetric matrix: A square matrix such that the element in the i-th row and j-th column is equal to the element in the j-th row and i-th column is called a symmetric matrix.
• Skew-symmetric matrix: A square matrix such that the element in the i-th row and j-th column is equal to the negative of the element in the j-th row and i-th column is called a skew-symmetric matrix.

Matrices - Transpose of a Matrix

• The transpose of a matrix is obtained by interchanging the rows and columns of the original matrix.
• The transpose of a matrix A is denoted as AT.
• If A = [aij] is an m x n matrix, then the transpose of A is an n x m matrix.
• The elements of the transpose matrix are obtained by interchanging corresponding elements of the original matrix.
• For example, if A = |1 2 3| |4 5 6| , then AT = |1 4| |2 5| |3 6|.

Matrices - Operations on Matrices

• Addition: Two matrices can be added if they have the same size. The sum of two matrices is obtained by adding corresponding elements of the matrices.
• Subtraction: Two matrices can be subtracted if they have the same size. The difference of two matrices is obtained by subtracting corresponding elements of the matrices.
• Scalar multiplication: A matrix can be multiplied by a scalar, which is a real number. The scalar multiplication of a matrix is obtained by multiplying each element of the matrix by the scalar.
• Matrix multiplication: Two matrices can be multiplied if the number of columns in the first matrix is equal to the number of rows in the second matrix. The product of two matrices is obtained by multiplying corresponding elements and summing the products.

Matrices - Properties of Matrix Operations

• Addition: The addition of matrices is commutative, i.e., A + B = B + A.
• Addition: The addition of matrices is associative, i.e., (A + B) + C = A + (B + C).
• Scalar multiplication: (k1 + k2)A = k1A + k2A, where k1 and k2 are scalars.
• Scalar multiplication: k1(k2A) = (k1k2)A, where k1 and k2 are scalars.
• Matrix multiplication: The matrix multiplication is associative, i.e., (AB)C = A(BC).
• Matrix multiplication: The matrix multiplication is distributive over addition, i.e., A(B + C) = AB + AC and (A + B)C = AC + BC.

</section> <section style="font-size: 50px";> <h2 id="determinants---definition-of-singular-matrix-1">Determinants - Definition of Singular Matrix</h2> <ul> <li>A matrix is said to be <strong>singular</strong> if its determinant is zero.</li> <li>The determinant of a matrix is denoted by |A| or det(A).</li> <li>If the determinant of a matrix is zero, it implies that the matrix is singular.</li> <li>A matrix is said to be <strong>non-singular</strong> if its determinant is non-zero.</li> <li>A non-singular matrix is also called an <strong>invertible</strong> or <strong>non-singular</strong> matrix.

</section> <section style="font-size: 50px";> <h2 id="determinants---evaluation-of-determinant-1">Determinants - Evaluation of Determinant</h2> <ul> <li>The determinant of a matrix can be evaluated using various methods.</li> <li>The most common methods are <strong>cofactor expansion</strong> and <strong>row reduction</strong>.</li> <li>Cofactor expansion method involves expanding the determinant along a row or column by using the formula det(A) = a1C1 + a2C2 + … + anCn, where ai is the element and Ci is its cofactor.</li> <li>Row reduction method involves simplifying the matrix by performing elementary row operations until the matrix is in a triangular or diagonal form.</li> <li>The determinant of a matrix is always a scalar quantity.

</section> <section style="font-size: 50px";> <h2 id="determinants---properties-of-determinants-1">Determinants - Properties of Determinants</h2> <ul> <li>Determinant of a matrix remains unaffected if its rows and corresponding columns are interchanged.</li> <li>The determinant of a matrix is zero if any two rows (or columns) are identical.</li> <li>If a matrix has one row (or column) consisting of all zeros, then its determinant is zero.</li> <li>If each element of a row (or column) of a matrix is multiplied by a scalar k, then the determinant of the new matrix is k times the determinant of the original matrix.</li> <li>If two rows (or columns) of a matrix are proportional, then its determinant is zero.

</section> <section style="font-size: 50px";> <h2 id="determinants---properties-of-determinants-contd">Determinants - Properties of Determinants (contd.)</h2> <ul> <li>If two rows (or columns) of a matrix are interchanged, then the determinant still remains the same in magnitude but changes its sign.</li> <li>If all the elements of a row (or column) of a matrix are multiplied by k, then the determinant of the new matrix is also multiplied by k.</li> <li>Determinants can be used to solve a system of linear equations.</li> <li>If the determinant of A is non-zero, then the system has a unique solution.</li> <li>If the determinant of A is zero, then the system either has infinitely many solutions or no solutions, depending on the consistency of the equations.

</section> <section style="font-size: 50px";> <h2 id="matrices---introduction-1">Matrices - Introduction</h2> <ul> <li>A matrix is a rectangular arrangement of numbers or symbols in rows and columns.</li> <li>The numbers or symbols in a matrix are called elements or entries.</li> <li>The size of a matrix is given by the number of rows and columns it contains.</li> <li>A matrix with m rows and n columns is said to be an m x n matrix, denoted as mxn matrix.</li> <li>A matrix with an equal number of rows and columns is called a <strong>square matrix</strong>.

</section> <section style="font-size: 50px";> <h2 id="matrices---types-of-matrices-1">Matrices - Types of Matrices</h2> <ul> <li><strong>Row matrix</strong>: A matrix with a single row and multiple columns is called a row matrix.</li> <li><strong>Column matrix</strong>: A matrix with a single column and multiple rows is called a column matrix.</li> <li><strong>Zero matrix</strong>: A matrix in which all elements are zero is called a zero matrix or null matrix.</li> <li><strong>Diagonal matrix</strong>: A square matrix in which all the non-diagonal elements are zero is called a diagonal matrix. The diagonal elements may or may not be zero.</li> <li><strong>Identity matrix</strong>: A square matrix in which all the diagonal elements are 1 and all the non-diagonal elements are zero is called an identity matrix, denoted by I.

</section> <section style="font-size: 50px";> <h2 id="matrices---types-of-matrices-contd">Matrices - Types of Matrices (contd.)</h2> <ul> <li><strong>Symmetric matrix</strong>: A square matrix such that the element in the i-th row and j-th column is equal to the element in the j-th row and i-th column is called a symmetric matrix.</li> <li><strong>Skew-symmetric matrix</strong>: A square matrix such that the element in the i-th row and j-th column is equal to the negative of the element in the j-th row and i-th column is called a skew-symmetric matrix.</li> <li>The elements of a matrix can be denoted by lowercase or uppercase letters.</li> <li>The transpose of a matrix is obtained by interchanging the rows and columns of the original matrix.</li> <li>The transpose of a matrix is denoted as AT.

</section> <section style="font-size: 50px";> <h2 id="matrices---transpose-of-a-matrix-1">Matrices - Transpose of a Matrix</h2> <ul> <li>If A = [aij] is an m x n matrix, then the transpose of A is an n x m matrix.</li> <li>The elements of the transpose matrix are obtained by interchanging corresponding elements of the original matrix.</li> <li>For example, if A = |1 2 3| |4 5 6| , then AT = |1 4| |2 5| |3 6|.

</section> <section style="font-size: 50px";> <h2 id="matrices---operations-on-matrices-1">Matrices - Operations on Matrices</h2> <ul> <li><strong>Addition</strong>: Two matrices can be added if they have the same size. The sum of two matrices is obtained by adding corresponding elements of the matrices.</li> <li><strong>Subtraction</strong>: Two matrices can be subtracted if they have the same size. The difference of two matrices is obtained by subtracting corresponding elements of the matrices.</li> <li><strong>Scalar multiplication</strong>: A matrix can be multiplied by a scalar, which is a real number. The scalar multiplication of a matrix is obtained by multiplying each element of the matrix by the scalar.</li> <li><strong>Matrix multiplication</strong>: Two matrices can be multiplied if the number of columns in the first matrix is equal to the number of rows in the second matrix. The product of two matrices is obtained by multiplying corresponding elements and summing the products.

Matrices - Operations on Matrices (contd.)

• Matrix multiplication: The matrix multiplication is not commutative, i.e., AB ≠ BA in general.
• Matrix multiplication: If A is an m x n matrix and B is an n x p matrix, then the product AB is an m x p matrix.
• Matrix multiplication: The number of columns in the first matrix must be equal to the number of rows in the second matrix for the multiplication to be defined.
• Matrix multiplication: The product of two matrices is obtained by multiplying corresponding elements of the row of the first matrix with the column of the second matrix and summing the products.
• Zero matrix: The product of any matrix A with the zero matrix of compatible size is the zero matrix.

Matrices - Inverse of a Matrix

• The inverse of a matrix is denoted as A^(-1).
• If A is a square matrix and there exists a matrix A^(-1) such that AA^(-1) = A^(-1)A = I, then A is called an invertible matrix or non-singular matrix.
• Only square matrices have inverses.
• A matrix is invertible if and only if its determinant is non-zero.
• If A is invertible, then (A^(-1))^(-1) = A.

Matrices - Properties of Inverse

• If A and B are invertible matrices of the same size, then (AB)^(-1) = B^(-1)A^(-1).
• If A is an invertible matrix, then (A^(-1))^T = (A^T)^(-1).
• If A is an invertible matrix, then kA is also invertible, where k is a scalar.
• If A is an invertible matrix, then (A^(-1))^(-1) = A.

Matrices - Elementary Row Operations

• Elementary row operations are operations performed on the rows of a matrix to obtain a row equivalent matrix.
• The three elementary row operations are:
  1. Interchange two rows.
  2. Multiply a row by a non-zero scalar.
  3. Add a multiple of one row to another row.
• Elementary row operations do not change the determinant of a matrix.
• The elementary row operations can be used to simplify a matrix and solve systems of linear equations.

Matrices - Row Echelon Form

• A matrix is said to be in row echelon form if:
  1. The first non-zero element in each row, called the leading entry, is 1.
  2. The leading entry of each row is to the right of the leading entry of the previous row.
  3. Any row containing all zeros is at the bottom.
• A matrix in row echelon form is useful in solving systems of linear equations.

Matrices - Reduced Row Echelon Form

• A matrix is said to be in reduced row echelon form if:
  1. It is in row echelon form.
  2. Each leading entry is the only non-zero entry in its column.
• A matrix in reduced row echelon form is unique for a given matrix and is also called the row-reduced echelon form or reduced row-echelon form (RREF).

Matrices - Gaussian Elimination

• Gaussian elimination is a method used to obtain the row echelon form of a matrix.
• The goal of Gaussian elimination is to simplify the matrix into a form that is easier to solve and understand.
• Gaussian elimination involves performing elementary row operations to transform the matrix into row echelon form.
• The final matrix obtained after Gaussian elimination can be used to solve systems of linear equations and evaluate determinants.

Matrices - Cramer’s Rule

• Cramer’s Rule is a method used to solve systems of linear equations by using determinants.
• Cramer’s Rule states that for a system of n linear equations in n variables, if the determinant of the coefficient matrix is non-zero, then the system has a unique solution.
• Cramer’s Rule involves evaluating the determinants of matrices formed by replacing each column of the coefficient matrix with the column matrix of constants.
• The solutions of the system can be obtained by dividing the determinants with the determinant of the coefficient matrix.

Matrices - Application of Matrices

• Matrices have various applications in different fields, including:
  1. Engineering: Matrices are used in solving systems of linear equations, engineering design, control systems, and signal processing.
  2. Computer Science: Matrices are used in computer graphics, image processing, artificial intelligence, and data analysis.
  3. Economics: Matrices are used in input-output analysis, utility theory, and mathematical modeling of economic systems.
  4. Physics: Matrices are used in quantum mechanics, matrix mechanics, and eigenvalue problems.
  5. Statistics: Matrices are used in multivariate analysis, regression analysis, and covariance matrices.

Conclusion

• Matrices and determinants are important topics in mathematics.
• Determinants help in understanding the singularity and non-singularity of a matrix.
• Matrices