 # Using Matrices to Solve a System of Equations. Multiplicative Identity Matrix The product of a square matrix A and its identity matrix I, on the left.

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Using Matrices to Solve a System of Equations

Multiplicative Identity Matrix The product of a square matrix A and its identity matrix I, on the left or the right, is A. AI = IA =A General Form: Must be a square matrix with “1”s along the diagonal with “0”s everywhere else

Example The identity matrix must be a square matrix (the same dimensions as the original) with 0’s in every cell except for 1’s in the main diagonal The same! Find the multiplicative identity of the following matrix:

Inverse Matrix How do we find the Inverse Matrix: The product of a square matrix A and its inverse matrix A -1, on the left or the right, is the identity matrix I. AA -1 = A -1 A =I (A Must be a square matrix) This is a difficult task. We will not calculate the inverse by hand. Instead we will use the calculator.

Converting a Systems of Equations to a Matrix Equation Identify all of the coefficients to the variables Make sure the equations are in alphabetical order Coefficient Matrix Variable Matrix Constant Matrix In order to solve this equation, we need to find the inverse of the Coefficient Matrix

Solving a Systems of Equations with Matrices Identify all of the coefficients to the variables Make sure the equations are in alphabetical order and that every variable is in each equation Coefficient Matrix “A” Variable Matrix “X” Constant Matrix “B” Solve: Convert the system of equations into a matrix equation:

Solving a Systems of Equations with Matrices AXB A -1 XB A XB A Multiply the inverse on both sides. But which order is correct? Multiply by the inverse of A to isolate the variable matrix Continued … OR 3x3. 3x1 3x1. 3x3

Solving a Systems of Equations with Matrices AXB A -1 XB A You do not need to calculate the inverse matrix! Multiply by the inverse of A to isolate the variable matrix Continued … Step 1: Write the matrix system by hand. Step 3: Enter this in your calculator to solve the system If… Then… THUS: Step 4: Write the Answer: Step 2: Store Matrix A and B in your calculator

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