Matrices and Systems of Equations

If m and n are positive integers, an m x n matrix (read “m x n”) is a rectangular array

In which each entry of the matrix is a real number. An m x n matrix has m rows and n columns.

Definition of Matrix

[ 𝑎11 𝑎12 𝑎13⋯𝑎21 𝑎22 𝑎23⋯𝑎31⋮

𝑎𝑚1

𝑎32⋮

𝑎𝑚 2

𝑎33⋯⋮

𝑎𝑚3⋯

𝑎1𝑛𝑎2𝑛𝑎3𝑛⋮

𝑎𝑚𝑛]

Determine the order of each matrix.

Matrix Order

Writing an Augmented Matrix

SolutionBegin by writing the linear system and aligning the variables.

(on board)

Writing an Augmented Matrix Continued

𝑁𝑒𝑥𝑡 ,𝑢𝑠𝑒 h𝑡 𝑒𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑠𝑎𝑛𝑑 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑡𝑒𝑟𝑚𝑠𝑎𝑠

Try this…

𝑊𝑟𝑖𝑡𝑒 h𝑡 𝑒𝑎𝑢𝑔𝑚𝑒𝑛𝑡𝑒𝑑𝑚𝑎𝑡𝑟𝑖𝑥 𝑓𝑜𝑟 h𝑡 𝑒

1. Interchange two rows 2. Multiply a row by a nonzero constant 3. Add a multiple of a row to another row.

Elementary Row Operations

Example

h𝐼𝑛𝑡𝑒𝑟𝑐 𝑎𝑛𝑔𝑒 h𝑡 𝑒 𝑓𝑖𝑟𝑠𝑡 𝑎𝑛𝑑 𝑠𝑒𝑐𝑜𝑛𝑑𝑟𝑜𝑤𝑠 𝑜𝑓 h𝑡 𝑒𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙𝑚𝑎𝑡𝑟𝑖𝑥 .[−1 2 00 1 32 −3 4

342 ]

Example

𝐴𝑑𝑑−2 𝑡𝑖𝑚𝑒𝑠 h𝑡 𝑒 𝑓𝑖𝑟𝑠𝑡 𝑟𝑜𝑤𝑜𝑓 h𝑡 𝑒𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙[1 2 −40 3 −20 −3 13

3−1−8]

Try this…

h𝐼𝑛𝑡𝑒𝑟𝑐 𝑎𝑛𝑔𝑒 h𝑡 𝑒 𝑓𝑖𝑟𝑠𝑡 𝑎𝑛𝑑 h𝑡 𝑖𝑟𝑑 𝑟𝑜𝑤𝑠𝑜𝑓 h𝑡 𝑒𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙𝑚𝑎𝑡𝑟𝑖𝑥 .

Try this…

𝐴𝑑𝑑−3 𝑡𝑖𝑚𝑒𝑠 h𝑡 𝑒 𝑓𝑖𝑟𝑠𝑡 𝑟𝑜𝑤𝑜𝑓 h𝑡 𝑒𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙𝑚𝑎𝑡𝑟𝑖𝑥 𝑡𝑜 h𝑡 𝑒

A matrix in row-echelon form has the following properties. Any rows consisting entirely of zeros occur at the bottom of

the matrix. For each row that does not consist entirely of zeros, the first

nonzero entry is 1 (called a leading 1). For two successive (nonzero) rows, the leading 1 in the

higher row is farther to the left than the leading 1 in the lower row.

A matrix in row-echelon form is in reduced row-echelon form if every column that has a leading 1 has zeros in every position above and below its leading 1.

Row-Echelon Form and Reduced Row-Echelon Form

Example

Row-Echelon Form

Reduced Row-Echelon Form

Determine whether each matrix is in row-echelon form. If it is, determine whether the matrix is in reduced row-echelon form.

Try this…

Write the augmented matrix of the system of linear equations.

Use elementary row operations to rewrite the augmented matrix in row-echelon form.

Write the system of linear equations corresponding to the matrix in row-echelon form and use back-substitution to find the solution.

Gaussian Elimination with Back-Substitution

Solve the system.

Example

Will be completed on board.

Solve the system.

Try this…