The Determinant of a Matrix



Inverse and Identitity Matrices

Teacher: Lead a class discussion using the following questions as a guide.

• Review Questions:

1) For real numbers, what is the additive identity?

2) For real numbers, what is the multiplicative identity?

3) For real numbers, what is the additive inverse?

4) For real numbers, what is the multiplicative inverse?

• So lets make some predictions…..

1) For a 2 X 2 matrix, what is the additive identity?

2) For a 3 X 3 matrix, what is the additive identity?

3) For a 4 X 4 matrix, what is the additive identity?

4) For an n X n matrix, what is the additive identity?

5) For a 2 X 2 matrix, what is the multiplicative identity?

2) For a 3 X 3 matrix, what is the multiplicative identity?

3) For a 4 X 4 matrix, what is the multiplicative identity?

4) For an n X n matrix, what is the multiplicative identity?

The Identity Matrix – A square matrix with ones along the main diagonal and zeros everywhere else.

[pic]

The identity matrix has the property that IA = AI = A for any matrix A.

Example: Let [pic] and verify that ID = DI = D.

Inverse of a Matrix

Let A be a square matrix with dimensions n X n. The inverse of A (denoted [pic]) is a square matrix such that

[pic].

A matrix has an inverse if the matrix is square and is non-singular (the determinant ( 0).

Example: Determine whether the following matrices have an inverse. If the matrix does not have an inverse, state why.

a) [pic] b) [pic] c) [pic]

Finding the Inverse of a 2 X 2 matrix

Let [pic] and suppose D = ad – cb ( 0. Then [pic] exists and is given by

[pic].

Example: Find the inverse of [pic] if it exists.

Example: Use your calculator to find the inverse of matrices below, if the inverse exists.

a) [pic]

b) [pic]

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