1. Matrix Determinants and Inverses
Lesson 12.3

4. How to Determine if Two Matrices are Inverses
Multiply the two matrices: AB and BA.
If the result is an identity matrix, then the matrices are inverses.
Example: Are A and B inverses?
                                   
No, their product does not equal the 2x2 identity matrix

5. Are C and D inverses?
Yes, their product equals the 3x3 identity matrix

6. Non-square matrices do not have inverses.
Inverse of a Matrix Multiplicative Inverse of a Matrix
For a square matrix A, the inverse is written A-1. When A is multiplied by A-1 the result is the identity matrix I.
Non-square matrices do not have inverses.
AA-1 = A-1A = I

7. Example:
For matrix A , its inverse is A-1
Since
AA-1 =
A-1A=

8. Requirements to have an Inverse The matrix must be square
(same number of rows and columns).
2. The determinant of the matrix must not be zero
A square matrix that has an inverse is called invertible or non-singular. A matrix that does not have an inverse is called singular.
A matrix does not have to have an inverse,
but if it does, the inverse is unique.

9. Can we find a matrix to multiply the first matrix by to get the identity?
Let A be an n n matrix. If there exists a matrix B such that AB = BA = I then we call this matrix the inverse of A and denote it A-1.

10. Check this answer by multiplying
Check this answer by multiplying. We should get the identity matrix if we’ve found the inverse.

11. Determinants

12. Finding the determinant of a matrix
= ad - bc
Determinants are similar to absolute values, and use the same notation, but they are not identical, and one of the differences is that determinants can indeed be negative.

13. NOTICE The difference is in the type of brackets
If you have a square matrix, its determinant is written by
taking the same grid of numbers and putting them inside
absolute-value bars instead of square brackets:
                    
If this is "the matrix A" (or "A")...
...then this is "the determinant of A" (or "det A").
     
NOTICE The difference is in the type of brackets

15. Evaluate the following determinant:
Multiply the diagonals, and subtract:   

16. Find the determinant of the following matrix:
Convert from a matrix to a determinant,
multiply along the diagonals,
subtract, and simplify:

17. The computations for 3×3 determinants are messier than for 2×2's.
Various methods can be used, but the simplest is probably
the following:   
Take a matrix A:
Write down its determinant:

18. Then multiply along the down-diagonals:
Extend the determinant's grid by rewriting the first two columns of numbers
Then multiply along the down-diagonals:

19. ...and along the up-diagonals

20. Add the down-diagonals and subtract the up-diagonals:

21. And simplify
Then det(A)= 1.

22. Find the determinant of the following matrix:
First convert from the matrix to its determinant, with the extra columns:

23. Then multiply down and up the diagonals:

24. Then add the down-diagonals, subtract the up-diagonals, and simplify for the final answer: