Week 3 - Inverses, Eigenvalues & eigenvectors

Question 1

2 ways to find out if an inverse of a matrix A exists

Answer 1

1. If the row-echelon form of A has n leading ones (from MA107)
2. if |A| =/= 0

A^-1 is UNIQUE so Ax=b will get a unique solution

Question 2

Formula to find an inverse using determinants for 2x2 matrices

Answer 2

A^-1 = 1/|A| (d -b)
(-c a)

Question 3

4-step method to find an inverse for 3x3 matrices

*quicker to use row ops for 4x4 and beyond

Answer 3

1. Find matrix of minors
2. ‘Sign’ <- cof(A)
3. Transpose matrix <- adj(A)
4. Divide by determinant to arrive at A^-1

Adjugate of A = transpose of the matrix of cofactors, cof(A)^T

Question 4

Cramer’s rule

(not very useful except for economists)

Answer 4

ONLY works if |A| =/= 0, so row ops is still the best way since guarantees a solution

To solve the matrix equation Ax=b, construct matrices Ai by replacing the ith column of A with b

xi = |Ai|/|A|

Question 5

How to find the eigenvalues of a square matrix A?

Answer 5

Find all numbers λ that satisfy the equation |A-λIn|=0

Question 6

For each eigenvalue of a square matrix A, how to find the eigenvectors?

How to check that the values are correct?

Answer 6

Find a BASIS of the solution space of (A-λIn)x=0

Must have Ax=λx and x =/= 0

Check Ax = λx

Question 7

What is an eigenvalue & eigenvector?

Answer 7

Ax = λx for some NON-ZERO VECTOR x

λ = eigenvalue of A (a number)
x = an eigenvector, for the eigenvalue λ, that satisfies the equation

Question 8

How to check whether your inverse, A^-1 is correct?

Answer 8

Check AA^-1 = In (identity matrix) or A^-1A = In

Question 9

What is a trick using eigenvalues?

Answer 9

Eigenvalues multiplied together = determinant of the matrix

(don’t forget about the multiplicity)