Matrices and Systems of Equations

Definition of Matrix 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. 𝑎11 𝑎12 𝑎13⋯ 𝑎21 𝑎22 𝑎23⋯ 𝑎31 ⋮ 𝑎𝑚1 𝑎32 ⋮ 𝑎𝑚2 𝑎33⋯ ⋮ 𝑎𝑚3⋯ 𝑎1𝑛 𝑎2𝑛 𝑎3𝑛 ⋮ 𝑎𝑚𝑛 TW provide further explanation and draw examples on the board of various matrix dimensions.

Matrix Order Determine the order of each matrix. 2 1 −3 0 0 0 0 0 5 0 2 −2 −7 4 TW use these problems during guided practice. Slide may be moved until the end of the presentation.

Writing an Augmented Matrix 𝑊𝑟𝑖𝑡𝑒 𝑡ℎ𝑒 𝑎𝑢𝑔𝑚𝑒𝑛𝑡𝑒𝑑 𝑚𝑎𝑡𝑟𝑖𝑐 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑠𝑦𝑠𝑡𝑒𝑚 𝑜𝑓 𝑙𝑖𝑛𝑒𝑎𝑟 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠. 𝑥+3𝑦=9 −𝑦+4𝑧=−2 𝑥−5𝑧=0 Solution Begin by writing the linear system and aligning the variables. (on board) SW copy the aligned version of the system from the board.

Writing an Augmented Matrix Continued 𝑁𝑒𝑥𝑡, 𝑢𝑠𝑒 𝑡ℎ𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑠 𝑎𝑛𝑑 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑡𝑒𝑟𝑚𝑠 𝑎𝑠 𝑡ℎ𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 𝑒𝑛𝑡𝑟𝑖𝑒𝑠. 𝐼𝑛𝑐𝑙𝑢𝑑𝑒 𝑧𝑒𝑟𝑜𝑠 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑚𝑖𝑠𝑠𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑠. 1 3 0⋮ 0 −1 4⋮ 1 0 −5⋮ 9 −2 0 TW point out that the coefficients from this matrix come from the system of equations from the previous slide.

Try this… 𝑊𝑟𝑖𝑡𝑒 𝑡ℎ𝑒 𝑎𝑢𝑔𝑚𝑒𝑛𝑡𝑒𝑑 𝑚𝑎𝑡𝑟𝑖𝑥 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑠𝑦𝑠𝑡𝑒𝑚 𝑜𝑓 𝑙𝑖𝑛𝑒𝑎𝑟 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠. 𝑥+𝑦+𝑧=2 2𝑥−𝑦+3𝑧=−1 −𝑥+2𝑦−𝑧=4 This slide will be used for guided practice.

Elementary Row Operations 1. Interchange two rows 2. Multiply a row by a nonzero constant 3. Add a multiple of a row to another row. TW provide examples of each of the operations listed on the slide. SW copy down these examples on their slides.

Example 𝐼𝑛𝑡𝑒𝑟𝑐ℎ𝑎𝑛𝑔𝑒 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑎𝑛𝑑 𝑠𝑒𝑐𝑜𝑛𝑑 𝑟𝑜𝑤𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑚𝑎𝑡𝑟𝑖𝑥. 0 1 3 −1 2 0 2 −3 4 4 3 2 TW provide an explanation of the operation that took place. −1 2 0 0 1 3 2 −3 4 3 4 2

Example 𝐴𝑑𝑑 −2 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑟𝑜𝑤 𝑜𝑓 𝑡ℎ𝑒 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑚𝑎𝑡𝑟𝑖𝑥 𝑡𝑜 𝑡ℎ𝑒 𝑡ℎ𝑖𝑟𝑑 𝑟𝑜𝑤. 1 2 −4 0 3 −2 2 1 5 3 −1 −2 TW show the steps that take place between the original matrix and the final matrix. 1 2 −4 0 3 −2 0 −3 13 3 −1 −8

Try this… 𝐼𝑛𝑡𝑒𝑟𝑐ℎ𝑎𝑛𝑔𝑒 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑎𝑛𝑑 𝑡ℎ𝑖𝑟𝑑 𝑟𝑜𝑤𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑚𝑎𝑡𝑟𝑖𝑥. 1 3 4 0 3 −1 1 2 −3 0 2 4 This slide will be used for guided practice.

Try this… 𝐴𝑑𝑑 −3 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑟𝑜𝑤 𝑜𝑓 𝑡ℎ𝑒 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑚𝑎𝑡𝑟𝑖𝑥 𝑡𝑜 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑟𝑜𝑤. 4 5 9 0 5 2 −2 7 −3 −1 3 1 This slide will be used for guided practice.

Row-Echelon Form and Reduced Row-Echelon Form 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.

Example 1 2 −1 0 1 0 0 0 1 4 3 −2 Row-Echelon Form 1 2 −1 0 1 0 0 0 1 4 3 −2 Row-Echelon Form 0 1 0 0 0 1 0 0 0 5 3 0 Reduced Row-Echelon Form TW describe specific reasons why each matrix is labeled. SW be required to copy down notes.

Try this… Determine whether each matrix is in row-echelon form. If it is, determine whether the matrix is in reduced row-echelon form. 1 0 −2 0 1 11 0 0 0 4 3 0 1 3 4 0 0 0 0 1 6 −2 0 −1 0 1 4 0 0 1 0 0 0 0 −6 0 This slide will be used for guided practice.

Gaussian Elimination with Back-Substitution 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.

Example Solve the system. 𝑥+ 𝑦+ 𝑧− 2𝑤 =−3 2𝑦− 𝑧 =2 2𝑥+ 𝑥− 4𝑦+ 𝑧− 3𝑤=−2 4𝑦− 7𝑧− 𝑤 =−19 Will be completed on board. TW complete the problem on the board. Students will copy down the necessary steps.

Try this… Solve the system. 𝑥+ 𝑦+ 𝑧− 2𝑤 =−3 2𝑦− 𝑧 =2 2𝑥+ 𝑥− 4𝑦+ 𝑧− 3𝑤=−2 4𝑦− 7𝑧− 𝑤 =−19 This slide will be used for guided practice.