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Linear Algebra for Team-Based Inquiry Learning: PREVIEW Edition — Instructor Version

Steven Clontz, Drew Lewis

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Section 4.3 Solving Systems with Matrix Inverses (MX3)

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Subsection 4.3.1 Warm Up

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Activity 4.3.1.

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Which of the following matrices is invertible? Find the inverse for the one that is invertible.
  1. [1−10−110101]
  2. [1−13−11−110−2]
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Subsection 4.3.2 Class Activities

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Activity 4.3.2.

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Consider the following linear system with a unique solution:
3x1−2x2−2x3−4x4=−72x1−x2−x3−x4=−1−x1+x3=−1−x2−2x4=−5
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(a)
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Suppose we let
T([x1x2x3x4])=[3x1−2x2−2x3−4x42x1−x2−x3−x4−x1+x3−x2−2x4].
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Which of these choices would help us solve the given system?
  1. Compute T([−7−1−1−5])
  2. Find [x1x2x3x4] where T([x1x2x3x4])=[−7−1−1−5]
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(b)
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How can we express this in terms of matrix multiplication?
  1. [3−2−2−42−1−1−1−10100−10−2][x1x2x3x4]=[−7−1−1−5]
  2. [x1x2x3x4]=[3−2−2−42−1−1−1−10100−10−2][−7−1−1−5]
  3. [x1x2x3x4][3−2−2−42−1−1−1−10100−10−2]=[−7−1−1−5]
  4. [x1x2x3x4]=[−7−1−1−5][3−2−2−42−1−1−1−10100−10−2]
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(c)
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How could a matrix equation of the form Ax→=b→ be solved for x→?
  1. Multiply: (RREFA)(Ax→)=(RREFA)b→
  2. Add: (RREFA)+Ax→=(RREFA)+b→
  3. Multiply: (A−1)(Ax→)=(A−1)b→
  4. Add: (A−1)+Ax→=(A−1)+b→
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Remark 4.3.3.

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The linear system described by the augmented matrix [A∣b→] has exactly the same solution set as the matrix equation Ax→=b→.
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When A is invertible, then we have both [A∣b→]∼[I∣x→] and x→=A−1b→, which can be seen as
Ax→=b→⇒A−1Ax→=A−1b→⇒x→=A−1b→
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Subsection 4.3.3 Individual Practice

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Activity 4.3.5.

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Solving linear systems using matrix multiplication is most useful when we are working with one common coefficient matrix, and varying the right-hand side. That is, when we have Ax→=b→ for several different values of b→.
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In the following, let A=[2−1−6213114] and consider the following questions about various equations of the form Ax→=b→?
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(a)
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Suppose that b→=[111]. If asked to solve the equation Ax→=b→, which of the following approaches do you prefer?
  1. Calculate RREF[A|b→].
  2. Calculate A−1 and then compute x→=A−1b→
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(b)
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Suppose that b→1,b→2,b→3=[111],[213],[−135]. If asked to solve each of the equations Ax→=b→1,Ax→=b→2,Ax→=b→3, which of the following approaches do you prefer?
  1. Calculate RREF[A|b→1], RREF[A|b→2], and RREF[A|b→3]
  2. Calculate A−1 and then compute x→=A−1b→1, x→=A−1b→2, and x→=A−1b→3
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(c)
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Suppose that b→1,…,b→10 are 10 distinct vectors. If asked to solve each of the equations Ax→=b→1,…,Ax→=b→10, which of the following approaches do you prefer?
  1. Calculate RREF[A|b→1], ... RREF[A|b→10].
  2. Calculate A−1 and then compute x→=A−1b→1, ... x→=A−1b→10.
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Subsection 4.3.4 Videos

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Video coming soon to this YouTube playlist
 1 
www.youtube.com/watch?v=kpOK7RhFEiQ&list=PLwXCBkIf7xBMo3zMnD7WVt39rANLlSdmj
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Exercises 4.3.5 Exercises

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Subsection 4.3.6 Mathematical Writing Explorations

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Exploration 4.3.6.

Use row reduction to find the inverse of the following general matrix. Give conditions on which this inverse exists.
[1bcdefghi]
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Exploration 4.3.7.

Assume that H is invertible, and that HG is the zero matrix. Prove that G must be the zero matrix. Would this still be true if H were not invertible?
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Exploration 4.3.8.

If H is invertible and r∈R, what is the inverse of rH?
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Exploration 4.3.9.

If H and G are invertible, is H−1+G−1=(H+G)−1?
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Exploration 4.3.10.

Prove that if A, P, and Q are invertible with PAQ=I, then A−1=QP.
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Subsection 4.3.7 Sample Problem and Solution

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