LINEAR ALGEBRA (Midterm Reviewer)

System of Linear Equations Matrices and Its Operations Gauss-Jordan Reduction The Inverse of a Matrix Definition and Properties of Determinant

Elimination Method

Substition Method

Elimination Method

Substition Method

Two Method of System Linear Equation?

Unique solution

Many solutions/Infinite

No solution.

Three System of linear equations are?

Determinant

-Is a number associated with every square matrix. It is a tool used in many branches of mathematics, science, and engineering.

basket-weave method

finding the determinants of a square matrix could only be used if the size of the matrix is 2 x 2 or 3 x 3.

co-factor expansion method

If the square matrix is more than 3 x 3, we will be

using other method known as the _____________.

Non singular and invertible

If there exists matrices ?

Matrix

rectangular array of numbers in the form

Elements

The numbers in the array are called the __________

Square Matrix

is an array of numbers having the same number of columns and rows, i.e. m = n.

Diagonal Matrix

is a square matrix for which every term off the main diagonal is zero.

Scalar Matrix

is a diagonal matrix for which all terms on the main diagonal are equal.

Equal Matrices

are matrices having the corresponding elements equal.

Zero Matrix

is a matrix having zero entries.

Identity Matrix

is a square matrix whose elements in the main diagonal are all one and for which every term off the main diagonal is zero.

Augmented matrix

is a matrix where constant is adjoining to the coefficient matrix.

Transpose matrix

Interchanging rows and columns of

a matrix

n and m

The letters stand for rows and columns

Gauss-Jordan Reduction

is a special method that involves systematically

eliminating variables from equations.

1. Write the linear system in the augmented matrix.

2. Obtain the reduced row echelon form of the augmented matrix by applying the elementary row operations.

3. The reduced row echelon form of the augmented matrix gives the solution to the linear system.

The following three procedures in solving the linear system by Gauss-Jordan Reduction are:

1: All rows consisting entirely of zeros are grouped at the bottom of the matrix.

2: The first nonzero element of each other row is 1 and called this as a leading entry of its row.

3: The leading entry 1 of each row after the first is positioned to the right of the leading entry 1 of the previous row.

4: Elements in a column that contains a leading entry 1 are all zeros.

A matrix is considered in reduced row echelon form if it satisfies the following properties.

Inverse Matrix

plays a very significant role in solving certain system of linear equations.

1. By definition of inverse of a matrix.

2. For any 2 x 2 matrix,

3. Gauss-Jordan Reduction for finding the inverse of a matrix.

There are 3 ways in finding the inverse of a matrix.

Singular matrix or Non-invertible

These linear systems have no solutions, so A has no inverse. Thus, A is a__________ or __________.

Singular matrix or Non-invertible

The reduced row echelon form cannot be of the form [In : A-1]. Then A -1 does not exist.

Systems of linear equations.

The students and the teachers can see to it that determinant gives them information about the solutions of _________________.

Matrices Operation

are useful in the applications of matrices.

Matrix Addition and Subtraction

Two or more matrices can be added and subtracted provided that those matrices have the same number of rows and the same number of columns, i.e., only matrices of the same size.

Matrix Multiplication

The product of the two matrices is possible if and only if the number of columns of the first matrix is exactly the same as the number of rows of the second. For example; the first matrix is a 2 x 3 and the second matrix is a 3 x 3, then the product of the first and the second matrices is a 2 x 3 matrix.

Scalar Multiplication

If r is any real number and A is an m x n matrix, then the scalar multiple of A by r is rA. Hence, rA is obtained by multiplying each element of A by r.