Accelerated Pre-Calculus Lesson 2 Essential Question: What is an inverse, how can I find the inverse of a matrix, and verify it is the inverse? How can I find the determinant of a square matrix? Section Objectives: Students will be able to find an inverse, and take a determinant. Standards: MCC9‐12.N.VM.10 (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. MCC9‐12.A.REI.9 (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
New Vocabulary Given a n x n matrix A, if there exists an inverse (A -1 ) of matrix A then A A -1 = A -1 A = I n If a square matrix A has NO inverse then it is called a singular matrix. If a square matrix has an inverse it is called a nonsingular matrix Cofactor expansion- Let A = [A ij ] be a matrix of order n x n where n > 2. The determinant of A denoted det(A) is the sum of the entries in any row or column multiplied by their respective cofactors.
Definition of an Inverse Given a n x n matrix A, if there exists an inverse (A -1 ) of matrix A then A A -1 = A -1 A = I n Example: A A -1 = 1 2 7 9 -1.5.5 1.25 -.25 1 0 0 1
If a square matrix A has NO inverse then it is called a singular matrix. If a square matrix has an inverse it is called a nonsingular matrix.
Example Verifying an inverse matrix Prove that A = and B = Are inverse matrices: 3-2 -1 1 1 2 1 3
How to find the inverse of a matrix by hand Let’s say we want to take the inverse of the following matrix We have to use the Gauss Jordan method. to do this rewrite the matrix with an identity matrix next to it. Our goal is to make the left side of the matrix look like the right side.
The only things you can do to a matrix to change its form is: 1. Interchange two rows. 2. Multiply a row by a nonzero constant. 3. Add a multiple of one row to another row.
We want the number in the first row and first column to be a 1. A 1 is already there, next we want all of the numbers below that 1 to be a zero.
What ever you are left with on the right is the inverse of the original matrix.
Finding a determinant of square matrices Determinant of a 2 x 2 If a 2 x 2 matrix A has an inverse then the determinant of A is as follows. A =, det A = ad - bc a b c d
Theorem An n x n matrix A has an inverse if and only if det A ≠ 0.
Finding a determinant of square matrices To take the determinant of a non 2 x 2 square matrix you have to do Cofactor expansion. Let A = [A ij ] be a matrix of order n x n where n > 2. The determinant of A denoted det(A) is the sum of the entries in any row or column multiplied by their respective cofactors.
Lets take the determinant of the following matrix 021 3-12 401 I have circled our cofactors
Using determinants to find the area of a triangle A triangle with vertices at (x 1, y 1 ), (x 2,y 2 ), and (x 3,y 3 ) Use: Area of the triangle = ½ base x height = ½ ( det ) X 1 y 1 1 X 2 y 2 1 X 3 y 3 1
Find the area of the following triangle The triangle with the following vertices: (4,0), (7,2), and (2,3)