Introduction and Definitions MATRIX Introduction and Definitions

Types of matrices Square matrix Diagonal matrix Scalar matrix Identity matrix (ones “1” on the main diagonal and zeros “0”everywhere else) Zero matrix (contains only zero elements) Negative matrix

Reduced row echelon form (RREF) Upper and lower triangular matrix (called upper-triangular if every element below the leading diagonal is zero and called lower-triangular matrix if every element above the leading diagonal is zero) Transpose matrix Symmetric matrix Skew symmetric matrix Row echelon form (REF) Reduced row echelon form (RREF)

Row Echelon Form (REF)

Reduced Row Echelon Form (RREF)

Determinants

Second Order Determinants Third Order Determinants Higher Order Determinants Minors and Cofactors

Second Order Determinants

Third Order Determinants

Note: Methods used to evaluate the determinant above is limited to only 2× 2 and 3×3 matrices. Matrices with higher order can be solved by using minor and cofactor methods.

Higher Order Determinants

Adjoint

The Inverse of a Square Matrix

Inverse of a Matrix Finding the inverse of a 2x2 Matrix Finding the inverse of a 3x3 or Higher order Matrix By using cofactor method By using elementary row operation

Finding the inverse of a 2x2 Matrix

ELEMENTARY ROW OPERATION INVERSE FOR 3X3 ELEMENTARY ROW OPERATION (ERO) COFACTOR METHOD NHAA/IMK/UNIMAP

Finding the inverse of a 3x3 or Higher order Matrix by using cofactor method

Find the inverse of each matrix using the Cofactor Method: Exercise: Find the inverse of each matrix using the Cofactor Method: NHAA/IMK/UNIMAP

Step 1: find the cofactor of A Answer : Solution for (a): Step 1: find the cofactor of A Step 2: find adj(A) NHAA/IMK/UNIMAP

Step 4: find the inverse of A Step 3: find det(A) Step 4: find the inverse of A NHAA/IMK/UNIMAP

Finding the inverse of a 3x3 or Higher order Matrix by using Elementary Row Operation

Inverse using Elementary Row Operations (ERO) Theorem 3 Let A and I both be nxn matrices, the augmented matrix may be reduced to by using elementary row operation (ERO) NHAA/IMK/UNIMAP

Inverse using Elementary Row Operations (ERO) Characteristics of ERO (i) : interchange the elements between ith row and jth row Example NHAA/IMK/UNIMAP

Inverse using Elementary Row Operations (ERO) Characteristics of ERO (ii) : multiply ith row by a nonzero scalar, k Example NEW R1 NHAA/IMK/UNIMAP

Inverse using Elementary Row Operations (ERO) Characteristics of ERO (iii) : add or subtract ith row to a constant multiple jth row by a nonzero scalar, k Example NEW R1 NHAA/IMK/UNIMAP

Types of Solutions to system Linear Equations

There are 3 possible solutions: 3 TYPES OF SOLUTIONS A SYSTEM WITH UNIQUE SOLUTION A SYSTEM WITH INFINITELY MANY SOLUTIONS A SYSTEM WITH NO SOLUTION NHAA/IMK/UNIMAP

A System with Unique Solution Consider the system: Augmented matrix: The system has unique solution: NHAA/IMK/UNIMAP

A System with Infinitely Many Solutions Consider the system: Augmented matrix: The system has many solutions: let where s is called a free variable. Then,

A system with No Solution Consider the system: Augmented matrix: The system has no solution, since coefficient of is ‘0’. NHAA/IMK/UNIMAP

Solving systems of Equations Inverses of Matrices Gaussian Elimination and Gauss-Jordan Elimination Cramer’s Rule

Inverses of Matrices

Inverses of Matrices

Gauss-Jordan Elimination Gaussian Elimination and Gauss-Jordan Elimination

Cramer’s Rule

Cramer’s Rule

Cramer’s Rule

Cramer’s Rule

Cramer’s Rule

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