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Reflections 7.7 Reflect on your understanding

In addition to the reflection activities below, re-read Section 7.2 Terminology and notation. Be sure you understand each of the new definitions introduced in this chapter, and spend some time committing them to memory.

1. Special form patterns.

Summarize the patterns in the entries of each of the special forms scalar, diagonal, upper triangular, lower triangular, and symmetric.

2. Special forms as coefficient matrices.

Consider each question below for each of the forms scalar, diagonal, upper triangular, and lower triangular.

(a)

What are the possibilities for the rank of a matrix in that form? What properties/patterns does the answer depend on?

(b)

How many solutions can a system have where the coefficient matrix has that form? What properties/patterns does the answer depend on?

3. RREF matrices.

(a)

True or false: Every square RREF matrix is upper triangular.

(b)

True or false: Every upper triangular matrix is in RREF.

4. Hierarchy of special forms.

Be sure you can articulate the reasoning behind your answer for each based on the definitions of the forms in the Terminology and notation section for this chapter.

(a)

True or false: Every diagonal matrix is also upper triangular.

(b)

True or false: Every diagonal matrix is also lower triangular.

(c)

True or false: A matrix that is both upper and lower triangular must be diagonal.

(d)

True or false: Every diagonal matrix is also symmetric.

(e)

True or false: Every scalar matrix is also diagonal.

(f)

True or false: Every scalar matrix is also upper triangular.

(g)

True or false: Every scalar matrix is also lower triangular.

(h)

True or false: Every scalar matrix is also symmetric.

5. Invertibility of special forms.

Remind yourself of the simple condition based on the entries of the matrix by which you can tell whether a scalar matrix is invertible. Repeat for diagonal and for triangular matrices.

6. Multiplying by diagonal.

State the pattern in the result of each of the following computation recipes.
  1. diagonal-matrix times matrix
  2. matrix times diagonal-matrix

7. Decomposing diagonal matrices.

Statement 7 of Theorem 6.5.2 says that every invertible matrix can be expressed as a product of elementary matrices. Describe a simple general pattern for how every invertible diagonal matrix can be expressed as a product of elementary matrices.
Hint.
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