In addition to the reflection activities below, re-read Section 4.2 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.Equality of matrices.
What condition(s) must be checked to verify that two matrices are equal?
2.Matrix operations.
(a)
For each, describe in one sentence (in words, not formulas!) how to carry out the operation.
matrix addition
matrix subtraction
scalar multiplication
transpose
(b)
Briefly describe (in words, not formulas!) how to carry out matrix multiplication. (This one might take more than one sentence.)
3.Systems via matrix equations.
Explain how the operation of matrix multiplication combined with the concept of matrix equality allows an entire system of linear equations to be represented as a single matrix equation \(A \uvec{x} = \uvec{b}\text{.}\)
4.Context of operation symbols and notation.
In this chapter we have introduced a number of new mathematical operations, but in most cases were are using old symbols and notation to represent these new operations. It is important to be able to use the context in which these symbols and notation appear to determine exactly what operation is represented.
Are the two plus signs in Rule 2.a the same type or different types?
(c)
Describe the various types of implied multiplication in the notation of Rule 2.d and Rule 2.e.
5.Domain of matrix multiplication.
(a)
What condition(s) must two matrices meet in order for their product to be defined?
(b)
Explain why both products \(A \utrans{A}\) and \(\utrans{A} A\) are always defined, for every matrix \(A\text{,}\) even if \(A\) is nonsquare.
6.Old rules no longer apply.
(a)
Explain why the commutative rule \(B A = A B\) is not true in general in matrix algebra.
(b)
Explain why the power rule \({(A B)}^m = A^m B^m\) (for \(m\) a positive integer) is not true in general in matrix algebra.
(c)
Explain why the binomial expansion rule \({(A + B)}^2 = A^2 + 2 A B + B^2\) is not true in general in matrix algebra.
(d)
Explain why the expression \(A X + B X\)cannot (in general) be factored as \(X (A + B)\) in matrix algebra. How can it be factored?
(e)
Can the expression \(A X + X B\) be factored in matrix algebra?
7.Number of solutions for a homogeneous systems.
(a)
Explain in terms of matrix algebra why it is true that every homogeneous system \(A \uvec{x} = \zerovec\) is consistent.
(b)
In the algebra of numbers we have the rule that if \(a b = 0\) then at least one of \(a = 0\) or \(b = 0\) must be true. What does the fact that a homogeneous system \(A \uvec{x} = \zerovec\) can have nontrivial solutions say about the possibility of such a rule in the algebra of matrices?
