2.5 DETERMINANTS AND MULTIPLICATIVE INVERSES OF MATRICESBy the end of the section students will evaluate determinants, find inverses of matrices, and set up matrix equations. Students will demonstrate this by creating a foldable.

How do we find a second order () determinant?

Notation:

The determinant helps us to find the inverse of a matrix, the determinant of a 2 by 2 is used in finding the determinant in a 3 by 3, which is what we will use when we get to VECTORS!!

*determinants are ONLY for square matrices

**Determinants that are = 0 are called “singular”

By the end of the section students will evaluate determinants, find inverses of matrices, and set up matrix equations. Students will demonstrate this by creating a foldable.

Example 1: Find the determinant of each second order matrixA.

B.

C.

By the end of the section students will evaluate determinants, find inverses of matrices, and set up matrix equations. Students will demonstrate this by creating a foldable.

Example 1: Find the determinant of each second order matrixD.

E.

F.

How do we find a third order () determinant?

Minor: the second order minor of is the matrix created when the row and column of are crossed out.

Hence the minor of is Recall:

How do we find a third order () determinant?

Notice the SIGNS in RED

By the end of the section students will evaluate determinants, find inverses of matrices, and set up matrix equations. Students will demonstrate this by creating a foldable.

Example 2: Find the determinant of each third order matrixA.

By the end of the section students will evaluate determinants, find inverses of matrices, and set up matrix equations. Students will demonstrate this by creating a foldable.

Example 2: Find the determinant of each third order matrixB.

What is an Inverse? (only for matrices)

• An inverse is something that UNDOES when solving an equation we use inverse operations

Is solved by using the inverse of “times 5” • We need to be able to do this for matrices…

Let , then

What happened to the first diagonal?What happened to the second diagonal?

What is an Inverse? (only for matrices)

What happened to the first diagonal?What happened to the second diagonal?

What happens if ?What was that called?

Example 3: Find the inverse of each second order matrixA.

B.

C.

By the end of the section students will evaluate determinants, find inverses of matrices, and set up matrix equations. Students will demonstrate this by creating a foldable.

Didn’t matrices come from systems of equations? When are we going to get to those because I LOVE solving systems of equations! More than Christmas!!

• Recall matrix multiplication

• Lets say we have the system of equations

We can write this as the matrix equationDÉJÀ VU!!!!

Example 4: Write the system of equations as a matrix equation and identify the matrix needed to solve the equation.

A. Matrix equation:

To solve we need to “undo” the matrix that is currently there, how do we “undo?”

INVERSE!!!

Inverse matrix to solve:

By the end of the section students will evaluate determinants, find inverses of matrices, and set up matrix equations. Students will demonstrate this by creating a foldable.

Example 4: Write the system of equations as a matrix equation and identify the matrix needed to solve the equation.

B. Matrix equation:

Inverse matrix to solve:

C.

By the end of the section students will evaluate determinants, find inverses of matrices, and set up matrix equations. Students will demonstrate this by creating a foldable.

Example 4: Write the system of equations as a matrix equation and identify the matrix needed to solve the equation.

C. Matrix equation:

Inverse matrix to solve:

What is this??

Lets look at the equations?

These are parallel lines and have NO solution.

By the end of the section students will evaluate determinants, find inverses of matrices, and set up matrix equations. Students will demonstrate this by creating a foldable.

Summary• Create your own four door foldable and create and

SOLVE a problem similar to each type of example given in class.

By the end of the section students will evaluate determinants, find inverses of matrices, and set up matrix equations. Students will demonstrate this by creating a foldable.