Question

4.) True or False : 3Z = Z + Z + Z when Z is a matrix.

􏰀1 2􏰁 􏰀a􏰁

5.) Let X = 3 4 ; E = b

Find each of the following. If the operation cannot be done : state undefined operation.

a) XE

b) EX

c) XT X where XT stands for the transpose of X

10.) Consider the linear equation 2a + 3b = 4

Is (a; b) = ( 1/2 ; 1) a solution to the equation? Motivate your answer.

11.) Look up what is meant by a system of linear equations.

A known fact of solutions of systems of linear equations is that only one the following options can hold :

(a) No solution possible

(b) A unique solution can be found

(c) The system has infinite solutions.

Consider that two straight lines form a linear system.

Interpret what happens geometrically to the straight lines to get each case of the solution types given above.

12.) Look up the concept of a homogeneous linear system.

Only two solution types of the three mentioned solution types above are possible. Which one can never happen and why.

Answer 1

4)

Let

$$Z=\begin{pmatrix} a_{11} & a_{12} & ... & a_{1m} \\ a_{21} & a_{22} & ... & a_{2m} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n1} & a_{n2} & ... & a_{nm} \\ \end{pmatrix}$$

Then

$$3Z=\begin{pmatrix} 3a_{11} & 3a_{12} & ... & 3a_{1m} \\ 3a_{21} & 3a_{22} & ... & 3a_{2m} \\ \vdots & \vdots & \vdots & \vdots \\ 3a_{n1} & 3a_{n2} & ... & 3a_{nm} \\ \end{pmatrix}$$

$$Z+Z+Z=\begin{pmatrix} a_{11} & a_{12} & ... & a_{1m} \\ a_{21} & a_{22} & ... & a_{2m} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n1} & a_{n2} & ... & a_{nm} \\ \end{pmatrix}$$

$$+\begin{pmatrix} a_{11} & a_{12} & ... & a_{1m} \\ a_{21} & a_{22} & ... & a_{2m} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n1} & a_{n2} & ... & a_{nm} \\ \end{pmatrix}+\begin{pmatrix} a_{11} & a_{12} & ... & a_{1m} \\ a_{21} & a_{22} & ... & a_{2m} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n1} & a_{n2} & ... & a_{nm} \\ \end{pmatrix}$$

$$=\begin{pmatrix} a_{11}+a_{11}+a_{11} & a_{12}+a_{12}+a_{12} & ... & a_{1m}+a_{1m}+a_{1m} \\ a_{21}+a_{21}+a_{21} & a_{22}+a_{22}+a_{22} & ... & a_{2m} +a_{2m}+a_{2m}\\ \vdots & \vdots & \vdots & \vdots \\ a_{n1}+a_{n1}+a_{n1} & a_{n2}+a_{n2}+a_{n2} & ... & a_{nm}+a_{nm}+a_{nm} \\ \end{pmatrix}$$

$$=\begin{pmatrix} 3a_{11} & 3a_{12} & ... & 3a_{1m} \\ 3a_{21} & 3a_{22} & ... & 3a_{2m} \\ \vdots & \vdots & \vdots & \vdots \\ 3a_{n1} & 3a_{n2} & ... & 3a_{nm} \\ \end{pmatrix}=3Z$$

$3Z = Z + Z + Z$ when $Z$ is a matrix is True.

5)

$$X=\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, E=\begin{pmatrix} a \\ b \end{pmatrix}$$

a)

$$XE=\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\begin{pmatrix} a \\ b \end{pmatrix}=\begin{pmatrix} a +2b \\ 3a+4b \end{pmatrix}$$

b)

The matrix $E$ is $2\times1$ matrix, the matrix $X$ is $2\times2$ matrix.

Since $1\not=2,$ then

$$EX=\begin{pmatrix} a \\ b \end{pmatrix}\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}=does\ not\ exist$$

c)

$$X^T=\begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$$

$$X^TX=\begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}=\begin{pmatrix} 1+9 & 2+12 \\ 2+12 & 4+16 \end{pmatrix}$$

$$=\begin{pmatrix} 10 & 14 \\ 14 & 20 \end{pmatrix}$$

10.) Consider the linear equation

$$2a + 3b = 4$$

If $(a, b)=(\dfrac{1}{2}, 1),$ then substitute

$$2(\dfrac{1}{2}) + 3(1) = 4$$

$$4=4, True$$

Therefore $(a, b)=(\dfrac{1}{2}, 1)$ is a solution to the equation $2a+3b=1.$

11)

a) Two lines are parallel lines or skew lines.

b) Two lines are intersecting lines.

c) Two lines are coincident lines.

12) Since a homogeneous system always has a solution (the trivial solution), it can never be inconsistent.

Therefore a homogeneous linear system can have:

(a) A unique solution.

Or

(c) Infinite solutions.