In addition to the reflection activities below, re-read Section 5.2 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.Identity matrices.
Explain the role that identity matrices play in matrix algebra.
2.Inverse matrices.
Explain the role that inverse matrices play in matrix algebra.
3.Fractions of matrices.
Explain why fraction notation should not be used in matrix algebra.
4.Inverse of inverses, products, and powers.
Assume that \(M,N\) are square and invertible matrices of the same size.
(a)
Explain the meaning of the notation \(\inv{(\inv{M})}\text{,}\) and give an explanation based on the definition of inverse of a matrix for why this formula must always compute to \(M\text{.}\)
(b)
Explain the meaning of the notation \(\inv{(M N)}\text{,}\) and give an explanation based on the definition of inverse of a matrix for why the result of this formula is (in general) equal to \(\inv{N} \inv{M}\) and not \(\inv{M} \inv{N}\text{.}\)
(c)
Let \(\ell\) represent an arbitrary positive integer. Explain the difference in the meaning of the notation \(\inv{(M^\ell)}\) versus the notation \({(\inv{M})}^\ell\text{,}\) and give an explanation based on the definition of inverse of a matrix for why the two formulas always compute to the same result.
5.Order of multiplication involving inverses.
Suppose \(A\) and \(P\) are square matrices of the same size, and that \(P\) is invertible. Can the expression \(\inv{P} A P\) be simplified to just \(A\) in all such cases? Explain why or why not.
6.Cancellation.
Suppose \(A,B,C\) are square matrices of the same size.
(a)
Is the cancellation rule
\begin{equation*}
A B = A C \qquad \implies \qquad B = C
\end{equation*}
always/sometimes/never valid? Pick one and explain.
(b)
Is the cancellation rule
\begin{equation*}
A B = C A \qquad \implies \qquad B = C
\end{equation*}
always/sometimes/never valid? Pick one and explain.
(c)
For each of Task a and Task b: if your answer is “sometimes,” under what conditions is it valid?
7.Systems with invertible coefficient matrix.
Consider a system \(A \uvec{x} = \uvec{b}\) where the coefficient matrix \(A\) is square and invertible.
(a)
Explain what conclusions can be drawn about the number of solutions to the system from the assumption that \(A\) is invertible.
(b)
What further can be said about the solution(s) in the case that the system is homogeneous (that is, \(\uvec{b} = \zerovec\))?
8.Invertibility of RREF matrices.
Is a square matrix that is in RREF always/sometimes/never invertible? Pick one and explain.
