Section 2.2 Matrix Methods for solving linear equations.
Running too close to late again this morning, but made it in on time.
Goals:
- Write the augmented matrix of a system of linear equations
- write the ystem from the augmented matrix
- Perform row operations on a matrix
- Solve systems of linear equations using matrices
- Express the solution of a system with an infinite number of solutions
A Matrix is a rectangular array of numbers enclosed with brackets. Each number in the matrix is an element of the matrix. Also referred to as an entry in the matrix. The size of a matrix is determined by the number of rows and columns in the matrix.
When building a matrix from a system of linear equations, each equation is represented as one row of the matrix. Each factor or variable of the equation is placed in its own column, and each column is reresentative of a single variable. Only the coefficients of the variables are actually placed in the matrix. Therefore, all the coefficients of x will be in one column, all the variables of y will be in a seperate column, and so one. Equations lacking a variable contained in other equations have a zero for that variable’s coefficient. All constants need to be in the final column, which represents values to the right of the equal sign.
Row operations on a matrix are used to solve systems of equations when the system is written as an augmented matrix. There are three basic row operations that can be performed while maintaining an equivalent matrix.
- Interchange any two rows.
- Multiply or divide all the elements of one row by a nonzero real number.
- Replace any row of the matrix with the sum of the elements of that row and a multiple of the elements of another row.
1 and 2 are fairly straightforward. 3 means that, where r1 and r2 are two rows of a matrix, r1 = (x)r2 + r1. Usually operation 3 is applied to cancel out elements of a row.
Row Echelon Form is the description of a matrix where:
- the entry in row 1, column 1 is a one, and every other entry in column 1 is a zero.
- The first nonzero entry in each row after the first row is a one, and every number beneath it in the same column is a zero
- Any row with zeros in every column but the final column is at the bottom of the matrix.
Rules 1 and 2 can really be conflated; rule 1 obeys rule 2. Rule 3 is likewise a natural result of following rule 2, and could be conflated with it.
Important to note is that the final column is the “equal” column; it contains the constants to the right of the equal sign, if the numbers were expressed as equations. Therefore, it doesn’t follow those rules. It is the only value directly influenced by the values of the other elements in its row, so if every other value in its row has a coefficient of zero, it must also have a value of zero, and would not be useful for determining results.
Work for next time: Try problems 1,3,7, 15, 13, 19 in section 2.2
Will collect 1, 7, 13, 19.