Matrices

Question 1

Dimensions of a matrix

Answer 1

Written as x X y

No. of rows x no. of columns

Question 2

Matrix vs column vector vs row vector

Answer 2

A matrix has 2+ rows and columns
A column vector has one column
A row vector has one row

Question 3

If the value of a variable is a matrix

Answer 3

Use bold, capital letters

Question 4

Adding/subtracting matrices

Answer 4

Must be the same dimensions

Add/subtract the elements in corresponding positions

Question 5

Scalar multiplication

Answer 5

Multiply everything in the matrix by the scalar

Question 6

Square matrix

Answer 6

Has the same amount of rows and columns

Question 7

Zero matrix

Answer 7

A matrix where all the elements are zero

Question 8

Identity matrix

Answer 8

A square matrix where the diagonal from the top left is filled with 1s, every other value is 0

Question 9

How to find an element in a multiplied matrix?

Answer 9

• Take the corresponding row from the first matrix
• Take the corresponding column from the second matrix
• Multiply first in row by first in column and so on
• Sum the answers

Question 10

What must be true to be able to multiply matrices

Answer 10

Columns in matrix 1 = rows in matrix 2

Question 11

Does AB = BA

Answer 11

No

Question 12

Dimensions of a multiplied matrix

Answer 12

Rows in a x columns in b

Question 13

Determinant

Answer 13

Effectively the inverse of a function, multiplying by the determinant is the same as multiplying by the original

Question 14

How to write determinant of matrix A

Answer 14

det(A) or |A|

Question 15

If det(A) = 0

Answer 15

A is a singular matrix and doesn’t have an inverse

Question 16

If det(A) is not equal to 0

Answer 16

A is non-singular and does have an inverse

Question 17

Determinant of matrix a b

c d

Answer 17

ad-bc

Question 18

Determinant of matrix a b c
d e f
g h i

Answer 18

a | e f | - b | d f | + c | d e |
| h i | | g i | | g h |

Do the 2x2 determinant of matrices not in the same row/column as the factor

Question 19

The minor of an element in a 3x3 matrix

Answer 19

The determinant of the matrix remaining when the row/column it is in are eliminated

Question 20

Inverse of a matrix

Answer 20

Written as M^-1, undoes the effect of a matrix

MM^-1 = M^-1M = I

Question 21

Inverse of a 2x2 matrix

Answer 21

1. ( d -b)
   ———-
   det(A). ( -c a)

Question 22

What must you do when multiplying by an inverse

Answer 22

Do it to the front of both sides of the equation or the back of both
Write that MM^-1 = I before removing

Question 23

A^T

Answer 23

Transpose of matrix A, inverted rows and columns

Question 24

Inverting a general matrix

Answer 24

1) find det(A)
2) make a matrix of minors
3) use that matrix with each second sign reversed
4) transpose it
5) multiply by 1/det(A)

Question 25

Using matrices to solve simultaneous equations

Answer 25

Write the coefficients of x.y and z in a matrix with the top row the first equation ensuring the order (x,y,z)
That multiplied by column vector (x y z) gives you the column vector of the answers
Inverse the 3x3 matrix
Multiply that inverse by the answers to find x.y,z
Write out and apply to context if necessary

Question 26

What can you do for matrix simultaneous equations?

Answer 26

Use your calculator as long as you write the output of each step

Question 27

Matrix simultaneous equations when you have to form your own

Answer 27

State what each variable is
Find the equations from the information, leave 0 as a coefficient if one only contains 2 variables
write as ax + by + cz = d
Repeat the steps from when you are given equations

Question 28

What do 3 simultaneous equations with 3 unknowns form?

Answer 28

3 planes in 3d

Question 29

Consistent

Answer 29

If there are solutions that satisfy all of the equations

Question 30

First step to find the geometric representation of an equation

Answer 30

If the coefficients and answer one equation are all those of another raised by a factor then those two are the same (consistent with infinitely many solutions)
If the coefficients are raised by the same factor but not the answer then those two are parallel (inconsistent - no solutions)
Else: continue

Question 31

Second check to find the geometric representation

Answer 31

Non-singular: one unique solution, consistent

Singular: inconsistent or infinitely many solutions, continue

Question 32

What to do for the geometric representation if singular

Answer 32

Eliminate one variable by substitution
If one is a multiple of the other it is a sheaf, consistent, infinitely many solutions
If not it is inconsistent with no solutions, a prism