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Section 4.2 The Inverse of a Matrix (MX2)

Subsection 4.2.1 Warm Up

Activity 4.2.1.

Consider the matrices:
A=[151032], B=[721103221113].
Without using technology, what is the third column of the product AB?

Subsection 4.2.2 Class Activities

Activity 4.2.2.

Let A=[271032111]. Find a 3×3 matrix B such that BA=A, that is,
[?????????][271032111]=[271032111]
Check your guess using technology.

Definition 4.2.3.

The identity matrix In (or just I when n is obvious from context) is the n×n matrix
In=[100010001].
It has a 1 on each diagonal element and a 0 in every other position.

Activity 4.2.5.

Let T:RnRm be a linear map with standard matrix A. Sort the following items into three groups of statements: a group that means T is injective, a group that means T is surjective, and a group that means T is bijective.
  1. T(x)=b has a solution for all bRm
  2. T(x)=b has a unique solution for all bRm
  3. T(x)=0 has a unique solution.
  4. The columns of A span Rm
  5. The columns of A are linearly independent
  6. The columns of A are a basis of Rm
  7. Every column of RREF(A) has a pivot
  8. Every row of RREF(A) has a pivot
  9. m=n and RREF(A)=I

Definition 4.2.6.

Let T:RnRn be a linear bijection with standard matrix A.
By item (B) from Activity 4.2.5 we may define an inverse map T1:RnRn that defines T1(b) as the unique solution x satisfying T(x)=b, that is, T(T1(b))=b.
Furthermore, let
A1=[T1(e1)T1(en)]
be the standard matrix for T1. We call A1 the inverse matrix of A, and we also say that A is an invertible matrix.

Activity 4.2.7.

Let T:R3R3 be the linear bijection given by the standard matrix A=[216213114].
(a)
To find x=T1(e1), we need to find the unique solution for T(x)=e1. Which of these linear systems can be used to find this solution?
  1. 2x11x26x3=x12x1+1x2+3x3=01x1+1x2+4x3=0
  2. 2x11x26x3=x12x1+1x2+3x3=x21x1+1x2+4x3=x3
  3. 2x11x26x3=12x1+1x2+3x3=01x1+1x2+4x3=0
  4. 2x11x26x3=12x1+1x2+3x3=11x1+1x2+4x3=1
(b)
Use that system to find the solution x=T1(e1) for T(x)=e1.
(c)
Similarly, solve T(x)=e2 to find T1(e2), and solve T(x)=e3 to find T1(e3).
(d)
Use these to write
A1=[T1(e1)T1(e2)T1(e3)],
the standard matrix for T1.

Activity 4.2.8.

Find the inverse A1 of the matrix
A=[0001101411041112]
by computing how it transforms each of the standard basis vectors for R4: T1(e1), T1(e2), T1(e3), and T1(e4).

Activity 4.2.9.

Is the matrix [231142055] invertible?
  1. Yes, because its transformation is a bijection.
  2. Yes, because its transformation is not a bijection.
  3. No, because its transformation is a bijection.
  4. No, because its transformation is not a bijection.

Observation 4.2.10.

An n×n matrix A is invertible if and only if RREF(A)=In.

Activity 4.2.11.

Let T:R2R2 be the bijective linear map defined by T([xy])=[2x3y3x+5y], with the inverse map T1([xy])=[5x+3y3x+2y].
(a)
Compute (T1T)([21]).
(b)
If A is the standard matrix for T and A1 is the standard matrix for T1, find the 2×2 matrix
A1A=[????].

Observation 4.2.12.

T1T=TT1 is the identity map for any bijective linear transformation T. Therefore A1A=AA1 equals the identity matrix I for any invertible matrix A.

Subsection 4.2.3 Individual Practice

Activity 4.2.13.

Now that we have defined the inverse of a matrix, we have the ability to solve matrix equations. In the following equations, A,B all denote square matrices of the same size and I denotes the identity matrix. For each equation, solve for X.

Subsection 4.2.4 Videos

Figure 43. Video: Invertible matrices
Figure 44. Video: Finding the inverse of a matrix

Exercises 4.2.5 Exercises

Subsection 4.2.6 Mathematical Writing Explorations

Exploration 4.2.14.

Assume A is an n×n matrix. Prove the following are equivalent. Some of these results you have proven previously.
  • A row reduces to the identity matrix.
  • For any choice of bRn, the system of equations represented by the augmented matrix [A|b] has a unique solution.
  • The columns of A are a linearly independent set.
  • The columns of A form a basis for Rn.
  • The rank of A is n.
  • The nullity of A is 0.
  • A is invertible.
  • The linear transformation T with standard matrix A is injective and surjective. Such a map is called an isomorphism.

Exploration 4.2.15.

  • Assume T is a square matrix, and T4 is the zero matrix. Prove that (IT)1=I+T+T2+T3. You will need to first prove a lemma that matrix multiplication distributes over matrix addition.
  • Generalize your result to the case where Tn is the zero matrix.

Subsection 4.2.7 Sample Problem and Solution

Sample problem Example B.1.19.