6- Matrix Inverse and Determinant

Question 1

What is a singular matrix?
What is a non singular matrix?
So a matrix is … if there is… such that…

Answer 1

A matrix is singular if it is invertible, otherwise it is non-singular.

It is invertible if there is a matrix A^-1 such that A-1A = I and AA^-1= I .

Question 2

A is invertible if and only if:

Answer 2

• is a square matrix
• has a determinant different from 0.

Question 3

A2x2:
- which is the determinant?
- which is the inverse? Remember the requirements

Answer 3

• detA= ad - bc
• the inverse is equal to 1/detA * |(d -b);(-c a)|

Question 4

Algorithm to find the matrix inverse:
- A is invertible if and only if…
- In this case what happens?
- Algorithm

Answer 4

• if it is row equivalent to In
• if there is a sequence of elementary row operations that reduces A to In then they also transform I to A^-1.
• this means that if the augmented matrix |A In| is row equivalent to |In A^1| then we found A^-1.

Question 5

Matrix inverse and Linear Equations:
Given an invertible matrix A, if we have Ax=b then x is equal to…

Answer 5

…A^-1*b

Question 6

What is the matrix transpose?

Answer 6

The transpose of a Matrix is the matrix A^t whose columns are formed from the corresponding rows of A.

Question 7

Properties of the matrix inverse:
- if A is invertible then what about A^-1…
- if Anxn and Bnxn are invertible then what about AB…
- if A is invertible then what about A^t…

Answer 7

• (A^-1)^-1 = A
• AB is also invertible and AB^-1 = (B^-1 * A^-1)
• A^t is invertible and (A^t)^-1 = (A^-1)^t

Question 8

Determinant of a matrix using the cofactor expansion:
- what is the (i,j) cofactor of a matrix A? Cij
- we make the cofactor expansion to calculate…
- 2 different types of cofactor expansion

Answer 8

• Cij = (-1)^(i+j) * detAij
• we calculate the determinant using the cofactor expansion;
• cofactor expansion across row x is the sum for j that goes from 1 to n of axj * Cxj;
• cofactor expansion across column y is the sum for i that goes from 1 to n of aiy * Ciy

Question 9

Determinant of a matrix using the gaussian elimination:
- we transform A to what?
- we set r equal to what?
- we set s equal to what?
- what is the value of the determinant? What can we say about the invertibility?

Answer 9

• A is transformed to the echelon form U;
• r is the number of row interchange that we performed;
• s is the product of the scaling values that we used;
• determinant is equal to (-1)^r * (product of pivots in U)/s if A is invertible. 0 if it is not invertible.