MAT1360 Classwork
MAT 2401 Discovery Lab 3.1 Names___________________________________
Objectives
• To determine when a matrix is invertible.
• To explore the concept of determinants.
• Look at the method of calculating determinants by minors.
• Use SageMath to facilitate the computation of determinants.
Instructions
• Do not look up any references including the textbook and internet.
• Do not use a calculator. You are supposed to learn SageMath.
• Use correct notations and do not skip steps.
• Two persons per group. Do not communicate with other groups.
Existence of Inverse Matrices
One of the important questions concerning the applications of matrices is whether a square matrix is invertible or not. To help us understand the situation, let us recall what we know about a [pic] square matrix.
In §2.3, we know for a [pic] matrix [pic], the inverse can be computed by the formula
[pic].
So, if [pic], the formula is undefined and therefore the inverse does not exist. The expression [pic] is called the determinant of the matrix [pic].
Notation
For the matrix [pic], the determinant of [pic] is denoted by either [pic] or [pic]. From the discussions above, we have,
[pic]. (1)
It is common to memorize the formula as “the subtraction of the products of the diagonal elements”
[pic].
1. (a) Is[pic] invertible?
| |
|[pic] |
| |
|Conclusion: Since [pic], |
(b) Is[pic] invertible?
| |
|[pic] |
| |
|Conclusion: |
(c) Find [pic] such that [pic] is not invertible.
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
For [pic] matrices with[pic], we do not have a formula for the inverses. To understand the principles behind the formulation of determinants are somewhat more difficult. We will demonstrate the ideas below with [pic] matrices. The same ideas generalized to all [pic] matrices.
The Determinant of a 3×3 Matrix
Let[pic]. Then
[pic]. (2)
It looks awful in the first sight. But we are going to discover the patterns hidden within it.
Observations
2. [pic] are the first two terms of formula (2).
(a) Factor out [pic] from the expressions. What is its relation with [pic]?
| |
| |
|[pic] |
| |
(b) Put boxes over [pic] and [pic] on the determinant notation.
[pic]
(c) [pic] is called a minor of [pic]. It is obtained by deleting the row and column containing [pic].
So, [pic] is the product of [pic] with its minor.
Do the second and third group of expressions in (2) have the same pattern? Let us find out on the next page.
3. (a) Factor out [pic] from the [pic]. What is its relation with [pic]?
| |
| |
|[pic] |
| |
(b) Put boxes over [pic] and [pic] on the determinant notation.
[pic]
(c) Again, [pic] is obtained by deleting the row and column containing [pic]. So it is the minor of [pic].
We see that the second group of expressions almost fit into the same pattern as the first group except a sign different.
So, [pic] is the product of [pic] with its minor.
Most books also consider this as the product of [pic] with the negative of its minor.
4. (a) Factor out [pic] from the [pic]. What is its relation with [pic]?
| |
| |
|[pic] |
| |
(b) Put boxes over [pic] and [pic] on the determinant notation.
[pic]
(c) Again, [pic] is obtained by deleting the row and column containing [pic]. So it is the minor of [pic].
We see that the third group of expressions fit into the exact same pattern as the first group.
So, [pic] is the product of [pic] with its minor.
Summary
In light of the observations above, formula (2) can be expressed as
[pic]. (3)
This is the common way how the formula is being understood and memorized. This is sometimes referred as “expanding by minors”. We will look at the obvious way to generalize this to bigger determinants after some practices with calculations.
5. (a) Compute [pic].
| |
|[pic] |
| |
| |
| |
| |
(b) To check your answer with SageMath, use the det() or determinant() commands.
| |
|A=Matrix([[1,2,3],[0,-1,1],[2,2,0]]); |
|show(A);show(A.det());show(A.determinant()); |
6. Compute [pic]. Check your answer with SageMath.
| |
|[pic] |
| |
| |
| |
| |
Generalizations
Larger determinant can be computed in the same fashion. The key to remember is to alternate the signs of the minors as demonstrated in the next example.
7. (a) Compute [pic] and [pic]
| |
|[pic] |
| |
|[pic] |
(b) Use the results of (a) to compute [pic]. Check your answer with SageMath.
| |
|[pic] |
| |
| |
| |
| |
| |
| |
Special Matrices
8. You can do this problem by hand or SageMath. No need to show details, just answers.
(a) Compute the following determinants.
| | | |
|[pic] |[pic] |[pic] |
| | | |
| | | |
(b) If [pic] is the [pic] zero matrix, then[pic].
(c) (a) Compute the following determinants.
| | | |
|[pic] |[pic] |[pic] |
| | | |
| | | |
(d) If [pic] is the [pic] identity matrix, then[pic].
(e) Compute the following determinants.
| | | |
|[pic] |[pic] |[pic] |
| | | |
| | | |
(d) If[pic]is the [pic] diagonal matrix [pic], then[pic].
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.