Matrix Determinants and InversesMatrix Determinants and Inverses

How to Determine if Two Matrices are Inverses 1.Multiply the two matrices: AB and BA. 2.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

Are C and D inverses?

Yes, their product equals the 3x3 identity matrix

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

For matrix A , its inverse is A-1

Example:

Since

AA-1 =

A-1A=

Requirements to have an Inverse1.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.

10

01

24

13?

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.

2

32

2

11

10

01

24

13

2

32

2

11

Can we find a matrix to multiply the first matrix by to get the identity?

72

31A

12

371A

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

10

011AA

= 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.

Finding the determinant of a matrix

If this is "the matrix A" (or "A")...

...then this is "the determinant

of A" (or "det A").

  

  

  

  

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:                     

NOTICE The difference is in the type of brackets

•Evaluate the following determinant:

Multiply the diagonals, and subtract:   

•Find the determinant of the following matrix:

Convert from a matrix to a determinant, multiply along the diagonals, subtract, and simplify:

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:

Extend the determinant's grid by rewriting the first two columns of numbers

 

Then multiply along the down-diagonals:

...and along the up-diagonals

 

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

Then det(A)= 1.

And simplify

Find the determinant of the following matrix:

 

First convert from the matrix to its determinant, with the extra columns:

 

Then multiply down and up the diagonals:

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

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