Discovery 7.1.
Carry out the following tasks for each of the special types of matrices defined above. Think in general, and consider every possible size of matrix, not just \(2\times 2\) and \(3\times 3\text{!}\) You don’t need to prove each answer, but you should be able to articulate an informal justification for each answer that doesn’t rely on examples (unless it’s a counterexample).
Tip.
When considering the questions in this activity for symmetric matrices, rather than trying to figure things out with examples, it is much easier to work algebraically with a letter \(A\) representing an arbitrary symmetric matrix, and use the definition of symmetric: \(\utrans{A} = A\text{.}\)
(a)
Write down both a \(2\times 2\) and a \(3\times 3\) example of the type. Is it clear why this type of matrix has been given its particular name?
(b)
Does the (square) zero matrix have this type? Does an identity matrix? Does every \(1\times 1\text{?}\)
(c)
If \(A\) is a matrix of this type, is every scalar multiple of \(A\) also of this type? Is \(\utrans{A}\) of this type?
(d)
If \(A\) and \(B\) are matrices of this type and of the same size, is their sum of this type? Their product? A power (with a positive exponent)?
(e)
[Omit this task for symmetric matrices.]
Recall that a matrix is invertible if and only if its RREF is the identity matrix. Based on this, can you come up with a simple condition by which you can determine whether a matrix of this type is invertible or not?
(f)
If \(A\) is an invertible matrix of this type, is its inverse also of this type?
Hint. for symmetric matrices
For the case of symmetric matrices, it will be too complicated to work by examples. Instead, consider the formula \(\utrans{(\inv{A})} = \inv{(\utrans{A})}\) from Proposition 5.5.8 and the definition of symmetric matrix above.
(g)
Come up with a condition or set of conditions on the entries \(a_{ij}\) of a square matrix \(A\) by which you can determine whether or not \(A\) is of this type.
Hint.
Here is an example of the type of condition we’re looking for, using the identity matrix: a square matrix \(A\) is equal to the identity matrix if \(a_{ii} = 1\) for all indices \(i\text{,}\) and \(a_{ij} = 0\) for all pairs of indices \(i,j\) with \(i\neq j\text{.}\)