1.6. Exercises

Write the $2\times 4$ matrix $A=(a_{ij})$ where $a_{ij}=2+i-2j$.

Let $A\in M_3(ℝ)$ with coefficients  $a_{ij}=2-i^2+j$ for all $1\le i,j\le 3$.

1. (i)

   Write down $A$.

2. (ii)

   Calculate $A^3$.

Calculate $-2A+3B$ where .

Let . Calculate the scalar product $v\cdot w$ and the products $Av$ and $Aw$.

Let $n\ge 1$ be an integer and $v,w\in {ℝ}^n$. Write $I_n$ for the identity $n\times n$ matrix. Prove that the scalar product

where $v^t\in M_{1\times n}(ℝ)$ is the transpose of $v$.

Calculate all possible products of pairs of elements (possibly equal) taken among the following matrices:

Calculate all the possible products of pairs of elements (possibly equal) taken among the following matrices:

Calculate all the products of two elements (possibly equal) taken among the following matrices:

Verify that

where

Let .

1. (i)

   Calculate $A^2,A^3,A^4,A^5$, and $A^6$.

2. (ii)

   Find the pattern, and state $A^n$ for all $n\in \mathbb{N}$.

Let .

1. (i)

   Find $A^2,A^3,A^4,A^5,A^6,$ and $A^7$.

2. (ii)

   What is the pattern? Give an expression for $A^n$ for all $n\in \mathbb{N}$.

Let $B\in M_4(ℝ)$ with coefficients 

1. (i)

   Write down $B$.

2. (ii)

   Calculate ${(B+B^t)}^2$.

3. (iii)

   Find the smallest positive integer $n$ such that $B^n$ is the zero matrix.

Let .

1. (i)

   Find the transpose $A^t$ of $A$.

2. (ii)

   Calculate $A+A^t$ and $A{A}^t$.

Let $A\in M_n(ℝ)$ for some $n\ge 1$.

1. (i)

   Prove that the matrix $(A+A^t)$ is symmetric, and that $(A-A^t)$ is skew-symmetric.

2. (ii)

   Find matrices $B$ and $C$ in $M_2(ℝ)$ such that $B$ is symmetric, $C$ is skew-symmetric and .

3. (iii)

   For an arbitrary matrix $A\in M_n(ℝ)$, find matrices $B$ and $C$ in $M_n(ℝ)$ such that $B$ is symmetric, $C$ is skew-symmetric and $A=B+C$.

The trace of a square matrix $A=(a_{ij})\in M_n(ℝ)$ is the sum

1. (i)

   Calculate the trace $tr(I_n)$ of the identity matrix of size $n$.

2. (ii)

   Let $A,B\in M_n(ℝ)$. Prove that $tr(AB)=tr(BA)$.

Do there exist matrices $A,B\in M_{2\times 2}(ℝ)$ such that $AB=0$ and all of the coefficients of $A$ and $B$ are non-zero? If so, give an example of such matrices. If not, then give a proof.

Find at least two matrices $A\in M_{2\times 2}(ℝ)$ which obey $A^2=(-1)I_2$, where $I_2$ is the $2\times 2$ identity matrix.

Assume $A$ is a $2\times 2$ matrix obeying $A^2=I_2$, such that all four of its coefficients are integers. One such matrix is $A=I_2$. Are there any other examples? Write down as many as you can. Can you prove you have found them all?

Given a smooth function $F:{ℝ}^n\to {ℝ}^m$, in multivariable calculus the Jacobian matrix of $F$ is the $m\times n$ matrix of partial derivatives: $J_F=(\partial F_i/\partial x_j)$. For example, if $F(x,y)=(x^2,x{y}^2)$ then

Write the Jacobian matrix of the following functions:

1. (i)

   $F(x,y)=(x^{3}y,\sin y)$  ($J_F\in M_{2\times 2}(ℝ)$)

2. (ii)

   $F(x,y,z)=xyz$  ($J_F\in M_{1\times 3}(ℝ)$)

3. (iii)

   $F(x)=(x,x^2,x^3)$  ($J_F\in M_{3\times 1}(ℝ)$)

If $F:{ℝ}^n\to {ℝ}^m$ and $G:{ℝ}^m\to {ℝ}^p$, then the chain rule in multivariable calculus says that the Jacobian matrix of $G\circ F$ at a point $P\in {ℝ}^n$ is the matrix product of the Jacobians of $G$ and $F$: $J_G(F(P))\cdot J_F(P)$. This generalises the usual chain rule of single-variable calculus.

If $F(x,y)=(x^2,y^3)$ and $G(x,y)=(xy,x+y)$, then use the multivariable chain rule to find the Jacobian matrix of $G\circ F$. Check your answer by directly computing the Jacobian matrix of $(G\circ F)(x,y)=(x^2{y}^3,x^2+y^3)$

Give an example of a square matrix $A$, of any size, such that $A^4\ne 0$, but $A^5=0$.

Let $A\in M_n(ℝ)$ be a square matrix. We can define the exponential of a matrix to be the infinite series

This infinite series always converges to some square matrix. Find the matrix $e^A$, when $A$ is each of the following matrices:

1. (i)

2. (ii)

3. (iii)

4. (iv)