A matrix is a rectangular array of numbers

A matrix is a rectangular array of numbers. The array is usually displayed within brackets. The dimensions of the matrix are the number of rows and columns in the array. Algebra 3 Section 3. 6 Solving Systems Using Matrices 3 columns 2 rows Matrix A has 2 rows and 3 columns and is a 2 x 3 matrix read “ 2 by 3”. It can be written as A or A 2 x 3 Each number in a matrix is a matrix element. A matrix element can be identified by its row and column numbers. In matrix A, a 12 is the element in row 1 and column 2. a 12 is the element 4. © Mr. Sims

Identify the indicated element. 1. ) a 32 = -2 2. ) a 22 =5 3. ) a 13 = 17 4. ) a 34 =0 © Mr. Sims

A system of equations can be represented efficiently with a matrix. Each matrix row represents an equation. The last matrix column shows the constants to the right of the equal signs. Each of the other columns shows the coefficients of one of the variables. system of equations 5 x x x-coefficients matrix + + 2 y 4 y = = 7 -8 y-coefficients vertical bar replaces equal sign and separates coefficients from constants © Mr. Sims

Represent the system of equations with a matrix: 2 x + y = 9 x – 6 y = -1 Represent the system of equations with a matrix: x – 3 y + z = 6 x + 3 z = 12 y = -5 x + 1 x – 3 y + z = 6 {no change} x + 0 y + 3 z = 12 {inserted 0 for missing y} 5 x + y + 0 z = 1 {added 5 x and inserted 0 for missing z} Write each equation in same variable order. Insert a zero coefficient where there is a missing variable. © Mr. Sims

Write the linear system of equations that the matrix represents: Each row shows [x coefficient, y-coefficient, constant] in that order. 5 x + 2 y = 7 {top row} 0 x + y = 9 {bottom row} 5 x + 2 y = 7 y=9 {simplified by removing 0 x from bottom row} Write the linear system of equations that the matrix represents: Each row shows: [x coefficient, y-coefficient, z-coefficient, constant] in that order. -2 x + 4 y + 9 z = 0 -x + 3 y + 0 z = 7 5 x + y + 9 z = 8 {first row} {second row} {third row} -2 x + 4 y + 9 z = 0 -x + 3 y = 7 5 x + y + 9 z = 8 {simplified by removing 0 z} © Mr. Sims

A matrix that represents a system of equations can be used to solve the system. In this way, you do not have to write the variables. To solve the system using the matrix, use the steps for solving by elimination. Each step is a row operation. Your goal is to use row operations to get a matrix in the form: {for a two by two matrix} or {for a three by three matrix} Notice the first matrix represents the system x = a and y = b, which will be the solution of a system of two equations in two unknowns. Also, the second matrix represents the system: x = a, y = b, and z = c © Mr. Sims

Row Operations Switch any two rows Multiply a row by a constant becomes Add one row to another becomes Combine any of these steps. © Mr. Sims

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