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Free Homework Math Help: Matrix Algebra

 

Matrix Algebra and Matrices

There is a branch of algebra called Linear Algebra that makes it possible to solve systems of linear equations by using matrices. A matrix is a rectangular array of numbers arranged in rows and columns sort of like a table. Each number in the matrix is called an element. The size of a matrix is described by the expression:

       m x n        where m is the number of rows and n is the number of columns.

Vectors

When a matrix only has one row or one column, it is called a vector.

Row Vector:       Column Vector:
[ 2, 3, 7, 4, 1, 3 ]              A matrix in linear algebra that has only one row or column is a vector

The following system of equations can be represented by a matrix and a vector.

       3x + 6y + 5z = 7
       2x + 5y + 7z = 8
       9x +   y - 3z = 2

A system of linear equations can be represented by a matrix and a vector

When a matrix is used to represent a system of linear equations, the coefficients of the variables become the elements. The resulting matrix is called a coefficient matrix. Sometimes, the constants are included in the last column and there may be a vertical line separating this column from the others. This type of matrix is called an augmented matrix.

A coefficient matrix represents the coefficients of a system of linear functions An augmented matrix in linear algebra includes constants as well as coefficients

Echelon Form

One of the ways of solving a matrix is by using echelon form ( also called triangular form ). The idea is to change the matrix so that it has the following properties:

  1. The rows with the most zeros are at the bottom of the matrix.
  2. The first nonzero number in each row starts with a one.
  3. The first nonzero number in a row starts in a column to the right of the first nonzero number in the row above.
Echelon form is one way of solving a matrix in math involving a system

Row Operations

To change a matrix into echelon form, operations are performed on the rows. The following row operations can be performed on a matrix:

  1. Any row can be interchanged with another row.
  2. All the elements in a row can be multiplied by the same number that is not equal to zero.
  3. Replace a row by its sum with the multiple of another row.

Row Operations Example

The following example shows how row operations are used to obtain echelon form in a matrix. The Gauss-Jordan method is used in this example.

1.Interchange is one type of matrix row operation

Interchange rows 2 and 3.

2.A row in a matrix can be multiplied and added or subtracted with another row

Multiply row 1 by 6.

3.In math, linear algebra is used to solve systems of linear equations

Subtract row 2 from row 1 and put the result in row 2.

R2 = 6R1 - R2

4.The goal of row operations is to achieve echelon form

Divide row 1 by 6.

5.In echelon form, the first row component is typically a one

Divide row 2 by 2.

6.A linear system can be solved with Gausian elimination

Multiply row 2 by 4 and row 3 by 7.

7.The Gauss-Jordan method is used on matrices of linear equations

Subtract row 3 from row 2 and put the result in row 3.

R3 = 4R2 - 7R3

8.An augmented matrix is also called a partitioned matrix

Divide row 2 by 4 and row 3 by 17

9.Matrices can be multiplied in different ways

Multiply row 3 by 2 and make the constant 5 in row 1 in the form of a fraction with 17 in the denominator.

10.Another way of solving a system in linear algebra is with the inverse matrix

Subtract row 3 from row 1 and put the result in row 1. Divide row 3 by 2.

R1 = R1 - 2R3

11.The rows or columns in a matrix can also be called vectors

Multiply row 3 by -6. Change the constant 9 in row 2 to a fraction with 17 in the denominator.

12.The dot product is one way to multiply matrices

Add row 3 to row 2 and put the result in row 2. Divide row 3 by -6.

R2 = R2 + (-6)R3

13.The cross product is another way to multiply with matrices

Multiply row 1 by 7 and row 2 by 3.

14.There are many ways to solve systems in math

Subtract row 1 from row 2 and put the result in row 1.

R1 = 3R2 - 7R1

15.Row operations can help solve math problems in linear algebra

Divide row 1 by -7 and row 2 by 7.