1.8. Matrices Exercises

Answer the following exercises based on the content from this chapter. The solutions can be found in the appendices.

Exercise 1.1

(a)   Write down the \(3 \times 3\) matrix \(A\) whose entries are given by \(a_{ij} = i+j.\)

(b)   Write down the \(4 \times 4\) matrix \(B\) whose entries are given by \(b_{ij} = (-1)^{i+j}.\)

(c)   Write down the \(4 \times 4\) matrix \(C\) whose entries are given by \( c_{ij} = \begin{cases} -1, & i>j, \\ 0, & i=j, \\ 1, & i<j. \\ \end{cases} \)

Exercise 1.2

The Hilbert matrix is the \(n \times n\) matrix \(H\) where the value of its elements are \(h_{ij} = \dfrac{1}{i + j - 1}\).

(a)   Write down the \(4 \times 4\) Hilbert matrix.

(b)   Show that an \(n \times n\) Hilbert matrix is symmetric.

Exercise 1.3

Given the matrices

\[\begin{split} \begin{align*} A &= \begin{pmatrix} 1 & -3 \\ 4 & 2 \end{pmatrix}, & B &= \begin{pmatrix} 3 & 0 \\ -1 & 5 \end{pmatrix}, \\ C &= \begin{pmatrix} 5 \\ 9 \end{pmatrix}, & D &= \begin{pmatrix} 1 & 1 & 3 \\ 4 & -2 & 3 \end{pmatrix}, \\ E &= \begin{pmatrix} 1 & 2 \\ 0 & 6 \\ -2 & 3 \end{pmatrix} & F &= \begin{pmatrix} 1 & -2 & 4 \end{pmatrix}, \\ G &= \begin{pmatrix} 4 & 2 & 3 \\ -2 & 6 & 0 \\ 0 & 7 & 1 \end{pmatrix}, & H &= \begin{pmatrix} 1 & 0 & 1 \\ 5 & 2 & -2 \\ 2 & -3 & 4 \end{pmatrix}. \end{align*} \end{split}\]

Calculate the following where possible:

(a)   \(A + B\)

(b)   \(B + C\)

(c)   \(A^\mathsf{T}\)

(d)   \(C^\mathsf{T}\)

(e)   \(3B - A\)

(f)   \((F^\mathsf{T})^\mathsf{T}\)

(g)   \(A^\mathsf{T} + B^\mathsf{T}\)

(h)   \((A + B)^\mathsf{T}\)

Exercise 1.4

Using the matrices from Exercise 1.3 calculate the following where possible:

(a)   \(AB\)

(b)   \(BA\)

(c)   \(AC\)

(d)   \(CA\)

(e)   \(C^\mathsf{T}C\)

(f)   \(CC^\mathsf{T}\)

(g)   \(DE\)

(h)   \(GH\)

(i)   \(A(DE)\)

(j)   \((AD)E\)

(k)   \(A^3\)

(l)   \(G^4\)

Exercise 1.5

Calculate the determinants of the square matrices from Exercise 1.3.

Exercise 1.6

For each non-singular matrix from Exercise 1.3 calculate its inverse. Show that your answers are correct.

Exercise 1.7

Show that \(AA^\mathsf{T}\) is a symmetric matrix. Hint: use the properties of matrix transpose.

Exercise 1.8

Show that \((AB)^{-1} = B^{-1}A^{-1}\). Hint: use the associativity law.

Exercise 1.9

If \(A\) and \(B\) are \(n \times n\) matrices is the following equation true?

\[(A + B)^2 = A^2 + 2AB + B^2\]

If not, under what conditions would it be true?

Exercise 1.10

An involutory matrix is a matrix that is its own inverse, i.e., it satisfies the equation \(A^2 = I\). Under what conditions is the following matrix an involutory matrix?

\[\begin{split}A = \begin{pmatrix} a & b \\ c & -a \end{pmatrix} \end{split}\]

Exercise 1.11

Which of the following statements are true? For the false statements, give one counter example where the statement doesn’t hold.

(a)   If \(A = B\) then \(AC = BC\).

(b)   If \(AC = BC\) then \(A = B\).

(c)   For \([O]_{ij} = 0\), if \(AB = O\) then \(A = O\) or \(B = O\).

(d)   If \(A + C = B + C\) then \(A = B\).

(e)   If \(A^2 = I\) then \(A = \pm I\).

(f)   If \(B = A^2\) and if \(A\) is an \(n \times n\) symmetric matrix then \(b_{ii} \geq 0\) for \(i = 1, 2, \ldots, n\).

(g)   If \(AB = C\) and if two of the matrices are square then so is the third.

(h)   If \(AB = C\) and if \(C\) has a single column then so does \(B\).

(i)   If \(A^2 = I\) then \(A^n = I\) for all integers \(n \geq 2\).

Exercise 1.12

Given the matrices

\[\begin{split} \begin{align*} A &= \begin{pmatrix} 1 & -3 \\ 4 & 2 \end{pmatrix}, & B &= \begin{pmatrix} 3 & 0 \\ -1 & 5 \end{pmatrix}, \end{align*} \end{split}\]

solve the following equations for \(X\).

(a)   \(5X = A\)

(b)   \(X + A = I\)

(c)   \(2X - B = A\)

(d)   \(XA = I\)

(e)   \(BX = A\)

(f)   \(A^2 = X\)

(g)   \(X^2 = B\)

(h)   \((X + A)B = I\)