Week 2 - Introduction to Inverse Matrices

Question 1

Week 2

Name two characteristics of an elementary matrix.

Answer 1

(1) Corresponds to a particular elementary row operation.
(2) These matrices are always invertible.

Question 2

What are the main three elementary row operations?

Answer 2

(1) Swapping two rows:
Ri <=> Rj
(2) Multiply a row by a non-zero constant:
Ri’ -> Ri
(3) Add a multiple of one row onto another:
Rj’ -> Rj + α Ri

Question 3

When is matrix A invertible?

Answer 3

If there is a matrix B such that AB and BA are both identities.

Question 4

What is the identity matrix? What happens when you multiply it with another matrix?

Answer 4

Pivot in every column, every other entry is zero.
Matrices multiplied by the identity is equivalent to multiplying by 1.

Question 5

Write this property as an expression:

(1) The inverse of A and then the inverse of that, you get back the original matrix A.

Answer 5

(A^-1)^-1 = A

Question 6

Write this property as an expression:

(2) Inverse of the product of A and B is the product of the inverses of A and B but in the opposite direction.

Answer 6

(A*B)^-1 = B^-1 A^-1

Question 7

Write this property as an expression:

(3) The inverse of A multiplied by scalar c, is the product of the inverses of c and A.

Answer 7

(c*A)^-1 = c^-1 A^-1

Question 8

Write this property as an expression:

(4) The transpose of the inverse of matrix A is the inverse of the transpose of matrix A.

Answer 8

(A^-1)^t = (A^t)^-1

Question 9

What is the proof for property 2?

(A*B)^-1 = B^-1 A^-1

Answer 9

= A B B^-1 A^-1

= A I A^-1

= A A^-1

= I

Question 10

How do you find the inverse of a square matrix?

Answer 10

Form augmented matrix (A | I ) and apply elementary row operations until the LHS is in RREF.

If you get (I | B), then the matrix is invertible –> A^-1 = B.

If not, then the matrix is not invertible.

Question 11

What are the 7 conditions that allow us to determine the invertibility of a matrix?

Answer 11

(1) A is row equivalent to I.
(2) A is invertible.
(3) Ax = 0 has only the trivial solution x = 0.
(4) Ax = b has exactly one solution for an b ∈ ℝn.
(5) The columns of A span ℝn.
(6) There is a matrix B with AB = I.
(7) There is a matrix B with BA = I.

Question 12

Explain how these conditions are dependent on each other:

(1) A is row equivalent to I.
(2) A is invertible.

Answer 12

Matrix A is only invertible if there is a matrix B such that AB and BA are both identities. This means that ERO’s need to be applied to acquire the identity matrix and this can only be achieved if the matrices are the same size. Therefore, it must be row equivalent.

Question 13

Explain how these conditions are dependent on each other:

(2) A is invertible.
(3) Ax = 0 has only the trivial solution x = 0.

Answer 13

x = Ix (sub in I = A^-1A)
x = A^-1Ax (sub in Ax = 0)
x = A^-10
x = 0

Question 14

Explain how these conditions are dependent on each other:

(3) Ax = 0 has only the trivial solution x = 0.
(1) A is row equivalent to I.

Answer 14

If A is not row equivalent to I then the RREF(A) will have a zero row and so fewer than n pivots –> non-trivial solution.

Question 15

Explain how these conditions are dependent on each other:

(4) Ax = b has exactly one solution for an b ∈ ℝn.
(5) The columns of A span ℝn (the spanning set of the columns).

Answer 15

Finding a solution to Ax = b corresponds to finding a way of writing b as a linear combination of the columns of A.

Question 16

When would matrix A be considered symmetric?

Answer 16

Matrix A is symmetric if it is equal to its own transpose: A^t = A. Only possible if it is a n x n square matrix.

Question 17

What does a transpose matrix mean?

Answer 17

It is the result of reflecting the matrix about the main diagonal, turning the rows into columns and the columns into rows.

Question 18

What are the 3 properties of n x n symmetric matrices?

Answer 18

(1) A^t is symmetric.
(2) A + B, A - B are symmetric.
(3) cA is symmetric (c ∈ ℝ).

Question 19

Prove that for any matrix C, both CC^t and C^tC are symmetric.

Answer 19

The condition for symmetric matrices is that it needs to be equal to its own transpose. So, to find out if the product of C and C transpose is symmetric, we need to take the transpose of that and see if it is equal. This will result in the product of the transposes but in the reverse order, and upon simplifying, we get back what we first started with.

Numerically, this can be written as:
(CC^t)^t = (C^t)^t C^t = CC^t

(Applies to C^tC as well).

Question 20

What is the upper triangular of a square matrix A?

Answer 20

If all entries below the main diagonal are zero (i.e. i > j –> aij = 0).

Note: i –> row entry position, j –> column entry position.

Note: a square matrix in RREF is upper triangular; our main objective when solving RREF is to have all values under the main diagonal to be zero before reducing the other side.

Question 20

What is the main diagonal of a square matrix A?

Answer 20

Consists of all those entries aij with i = j.

Note: i –> row entry position, j –> column entry position.

Question 21

What is the lower triangular of a square matrix A?

Answer 21

If all entries above the main diagonal are zero (i.e. i < j –> aij = 0).

Note: i –> row entry position, j –> column entry position.

Question 22

What is a unit upper/lower triangular matrix?

Answer 22

With 1’s on the main diagonal.

Question 23

What happens when you take the transpose of an upper/lower triangular matrix?

Answer 23

A matrix A is upper triangular if A^t is lower triangular, and vice versa.

Question 24

What is LU factorisation of a square matrix A?

Answer 24

It consists of a unit lower triangular matrix L and a unit upper triangular matrix U such that A = LU (written as a product).

Question 25

Evaluate the LU factorisation formula A = LU.

Answer 25

If A can be reduced to REF using only ERO Rj’ -> Rj + α Ri with j < i, we obtain the upper triangular matrix U.

This can be written as a product of A and the series of ERO’s that were applied.
Er…E1A = U

Elementary matrices that correspond to these ERO’s are invertible, which means we can take it to the other side (multiplied in the reverse order).
A = E1^-1…Er^-1U

The inverse of an upper triangular matrix is a lower triangular matrix. Hence,
A = LU

where,
L = E1^-1…Er^-1
U is REF

Question 26

How do you solve equations using LU factorisations?

Answer 26

To solve Ax = b, form an LU factorisation A = LU, then sub that into the equation.

LUx = b (since Ux = y).
Ly = b (use to solve for y).

Ux = y (use y to solve for x).

Then, Ax = LUx = Ly = b.

Question 27

What is the definition of a determinant? How is it denoted?

Answer 27

It is a scalar value that is a function of the entries of a square matrix. It is commonly denoted det(A) or |A|.

Question 28

How is the determinant calculated for 2x2 or 3x3 matrices?

Answer 28

2 x 2 square matrix: A = (a b);(c d)
|A| = ad - bc

3 x 3 square matrix: A = (a b c; d e f ; g h i)
|A| = a (e f ; h i) - b (d f; g i) + c (d e; g h)

Question 29

What is the general formula for calculating the determinant of any size square matrix?

Answer 29

|A| = a11|A11| - a12|A12|+….+(-1)^n a1n|A1n|

Note: all the numbers are subscript form and n is the size of the matrix.

Question 30

What does the determinant tell us about invertibility of a matrix?

Answer 30

If det(A) is a non-zero number, then A will be invertible.

If det(A) = 0, then A is not invertible.

Question 31

How do you calculate the determinant if matrix A is either in upper triangular or lower triangular?

Answer 31

|A| is the product of the diagonal entries.

Question 32

Matrix B is obtained from matrix A by swapping two rows. What happens to the determinant of B?

Answer 32

|B| = - |A|
Changes the sign of the determinant.

Question 33

Matrix B is obtained from matrix A by multiplying one row by a non-zero scalar c. What happens to the determinant of B?

Answer 33

|B| = c|A|
Multiply the determinant of A by the non-zero scalar c.

Question 34

Matrix B is obtained from matrix A by adding a multiple of one row onto another row. What happens to the determinant of B?

Answer 34

|B| = |A|
Determinant is unchanged.

Question 35

What happens to the determinant of A when it is transposed?

Answer 35

|A| = |A^t|
Determinant is unchanged.

Question 36

What is the determinant of AB?

Answer 36

|AB| = |A||B|
Product of the determinants.

Question 37

What is the determinant of A^-1?

Answer 37

|A^-1| = |A|
Determinant is unchanged.