Matrix Determinants and Inverses

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Lesson 12.3

• Matrix Determinants and Inverses

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• 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
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Are C and D inverses?
Yes, their product equals the 3x3 identity matrix
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Inverse of a MatrixMultiplicative 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
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Example
For matrix A , its inverse is A-1
Since
AA-1
A-1A
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• 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.

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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.
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Check this answer by multiplying. We should get
the identity matrix if weve found the inverse.
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Determinants
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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.
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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
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• Evaluate the following determinant

Multiply the diagonals, and subtract   
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• Find the determinant of the following matrix

Convert from a matrix to a determinant,
multiply along the diagonals, subtract, and
simplify
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The computations for 33 determinants are messier
than for 22's. Various methods can be used, but
the simplest is probably the following   
Take a matrix A
Write down its determinant
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Extend the determinant's grid by rewriting the
first two columns of numbers
Then multiply along the down-diagonals
 
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...and along the up-diagonals
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Add the down-diagonals and subtract the
up-diagonals
 
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And simplify
Then det(A) 1.
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Find the determinant of the following matrix
First convert from the matrix to its determinant,
with the extra columns
 
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Then multiply down and up the diagonals
 
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Then add the down-diagonals, subtract the
up-diagonals, and simplify for the final answer

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http//www.purplemath.com/modules/determs.htm