Pure 1.6 - Matrices

Question 1

What is a matrix?

Answer 1

A matrix is an array of elements set out in a pair of brackets

Question 2

How can the size of a matrix described?

Answer 2

By saying the number of rows and columns, for example, a 2x2 matrix is one with 2 rows and 2 columns and a 2x4 matrix is one with 2 rows and 4 columns

Question 3

What is a square matrix?

Answer 3

One where the number of rows is the same as the number of columns

Question 4

What is a zero matrix?

Answer 4

One in which all of the elements are zero

The zero matrix is denoted by 𝟎

Question 5

What is an identity matrix?

Answer 5

A square matrix in which the elements on the leading diagonal (top left to bottom right) are 1 and all the remaining elements are 0
Identity matrices are denoted by 𝐈ₖ, where k represents size of the matrix

Question 6

How would you add or subtract matrices?

Answer 6

You add or subtract the corresponding elements in each matrix
Only matrices of the same size may be added or subtracted and those which are the same size are said to be additively conformable

Question 7

What notation is used normally to represent matrices?

Answer 7

Normally a bold capital letter such as 𝐀 or 𝐁

Question 8

How would you multiply or divide a matrix by a scalar?

Answer 8

Multiply or divide each element within the matrix by the scalar

Question 9

What is a scalar?

Answer 9

A number that isn’t a vector or a matrix, they are represented by non-bold letters and numbers

Question 10

How would you multiply matrices together?

Answer 10

To find the product of two multiplicatively conformable matrices, you multiply the elements in each row in the left-hand matrix by the corresponding elements in each column in the right hand matrix, then add the results together`

Question 11

What does it mean for a matrix to be multiplicatively conformable with another matrix?

Answer 11

If a matrix is multiplicatively conformable with another matrix, the two can be multiplied together
Matrices are multiplicatively conformable if the number of columns in the first matrix is the the same as the number of rows in the second matrix

Question 12

What is a product matrix?

Answer 12

The product of the multiplication of two matrices

It will have the same number of rows as the first matrix and the same number of columns as the second matrix

Question 13

Describe the importance of the order that you multiply matrices together in.

Answer 13

𝐀𝐁=𝐂
However it may be that 𝐁𝐀≠𝐂
In general 𝐀𝐁≠𝐁𝐀
And, if 𝐀𝐁 exists, 𝐁𝐀 doesn’t necessarily exist

Question 14

What is a determinant?

Answer 14

A scalar value associated with that matrix, the determinant of matrix 𝐌 can be written as det𝐌 or |𝐌| (Also written as replacing the bracket’s around 𝐌’s elements with |’s)
It is also sometimes written Δ

Question 15

How would you calculate the determinant of a 2x2 matrix, 𝐌?
Where 𝐌=
(𝑎 𝑏)
(𝑐 𝑑)

Answer 15

The determinant of 𝐌 is 𝑎𝑑-𝑐𝑏

Question 16

How would you calculate the determinant of a 3x3 matrix, 𝐌?
Where 𝐌=
(𝑎 𝑏 𝑐)
(𝑑 𝑒 𝑓)
(𝑔 𝘩 𝑖)

Answer 16

Reduce the 3x3 determinant to 2x2 determinants using the formula:
|𝑎 𝑏 𝑐|__|𝑒 𝑓|_ |𝑑 𝑓|__|𝑑 𝑒|
|𝑑 𝑒 𝑓|=𝑎|𝘩 𝑖 |-𝑏|𝑔 𝑖 |+𝑐|𝑔 𝘩|
|𝑔 𝘩 𝑖 |

Question 17

What does it mean for the determinant of a matrix to be 0?

Answer 17

The matrix is singular

Question 18

What does it mean for the determinant of a matrix to not be 0?

Answer 18

The matrix is non-singular

Question 19

What is the inverse of a matrix?

Answer 19

The inverse of a matrix 𝐌, written 𝐌⁻¹, is a matrix such that 𝐌𝐌⁻¹=𝐌⁻¹𝐌=𝐈

Question 20

How would you invert a 2x2 matrix, 𝐌?
Where 𝐌=
(𝑎 𝑏)
(𝑐 𝑑)

Answer 20

𝐌⁻¹= (1/det𝐌)(𝑑 -𝑏)

__________(-𝑐 𝑎 )

Question 21

What does it mean if a matrix is non-singular?

Answer 21

The inverse of the matrix can be found

Question 22

For two non-singular, multiplicatively conformable matrices, 𝐀 and 𝐁, how would you find (𝐀𝐁)⁻¹?

Answer 22

𝐀⁻¹𝐁⁻¹

Question 23

What is the transpose of a matrix?

Answer 23

The transpose of a matrix is the matrix found by interchanging the rows and the columns
The transpose of a matrix 𝐌 is written 𝐌ᵀ

Question 24

How would you find the inverse of a 3x3 matrix, 𝐀

Answer 24

1) Find the determinant of the matrix
2) Form the matrix of minors of the matrix
3) Form the matrix of cofactors by changing the signs of relevant elements according to the rule of alternating signs as illustrated:
(+ - +)
(- + - )
(+ - +)
4) Write down the transpose of the matrix of cofactors, 𝐂ᵀ
5)The inverse of the matrix is given by the formula
𝐀⁻¹=(1/det𝐀)𝐂ᵀ

Question 25

How can a matrix be used to solve a system of three simultaneous linear equation in three unknowns?

Answer 25

If 𝐀(𝑥)=𝒗, then (𝑥)=𝐀⁻¹𝒗
\_\_\_(𝑦) \_\_\_\_\_\_(𝑦)
\_\_\_(𝑧) \_\_\_\_\_\_(𝑧)
If 𝐀 is non-singular, a unique solution for
(𝑥)
(𝑦)
(𝑧)
can be found for any vector 𝒗

Question 26

What does it mean if a system of linear equations is consistent?

Answer 26

A system of linear equation is consistent if there’s at least one set of values that satisfies all the equations simultaneously
Otherwise, it is inconsistent

Question 27

If a set of planes are formed from a system of three linear equations, what do the meeting points between these planes show?

Answer 27

If the planes meet at a point, the system is consistent and has exactly one set solution represented by this point
If the planes form a sheaf, the system is consistent, with infinitely many solutions represented by the line of intersection of the three planes
If the planes form a prism, the system is inconsistent and has no solutions
If two or more planes are parallel then the system is inconsistent and has no solutions
If all three equations can be represented by the same planes, then the system is consistent with infinitely many solutions