1.8. Matrices Exercises

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

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 1.3

Using the matrices from Exercise 1.2 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 1.4

Using the matrices from Exercise 1.2 calculate the following:

(a)   \(\det(A)\)

(b)   \(|B|\)

(c)   \(\det(3A)\)

(d)   \(\det(G)\)

(e)   \(\operatorname{adj}(B)\)

(f)   \(\operatorname{adj}(H)\)

(g)   \(A^{-1}\)

(h)   \(B^{-1}\)

(i)   \(G^{-1}\)

(j)   \((AB)^{-1}\)

(k)   \(B^{-1}A^{-1}\)

(l)   \((DE)^{-1}\)

Exercise 1.5

Using the properties of determinants and solutions from Exercise 1.4 where necessary, find the determinants for the following matrices.

(a)   \(\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}\)

(b)   \(\begin{pmatrix} 4 & 2 \\ 1 & -3 \end{pmatrix}\)

(c)   \(\begin{pmatrix} 1 & 0 \\ -1 & 0 \end{pmatrix}\)

(d)   \(\begin{pmatrix} 4 & 2 & 3 \\ -4 & 12 & 0 \\ 0 & 7 & 1 \end{pmatrix}\)

(e)   \(\begin{pmatrix} 1 & 2 & 1 \\ -3 & -6 & 1 \\ 2 & 4 & 4 \end{pmatrix}\)

(f)   \(\begin{pmatrix} 3 & 6 \\ -1 & 3 \end{pmatrix}\)

Exercise 1.6

Prove that adding a multiple of a row or column to another row or column does not change the value of the determinant for a \(2\times 2\) matrix.

Exercise 1.7

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)   \((X + A)B = I\)