Matrices

Question 1

How many rows does a 3x2 matrix have?

Answer 1

3

Question 2

How many columns does a 3x2 matrix have?

Answer 2

2

Question 3

How many rows does a 2x3 matrix have?

Answer 3

2

Question 4

How many columns does a 2x3 matrix have?

Answer 4

3

Question 5

Does AB = BA? Why?

Answer 5

No, because matrices are not commutative

Question 6

State the 2x2 identity matrix

Answer 6

( 1 0 )
( 0 1 )

Question 7

State the 3x3 identity matrix

Answer 7

( 1 0 0 )
( 0 1 0 )
( 0 0 1 )

Question 8

Can you multiply a 3x2 matrix by a 2x1 matrix?

Answer 8

Yes

Question 9

Can you multiply a 3x2 matrix by a 1x2 matrix?

Answer 9

No

Question 10

How do you calculate the determinant of a 2x2 matrix?

Answer 10

△ = ad - bc

Question 11

What is a singular matrix?

Answer 11

A matrix where the determinant is 0

Question 12

What is a non singular matrix?

Answer 12

A matrix where the determinant is not 0

Question 13

What is a minor of a matrix?

Answer 13

A section of a matrix used to find the determinant of a 3x3 matrix

Question 14

How do you calculate the determinant of a 3x3 matrix?

Answer 14

• Find the minors of the first 3 elements of the matrix
• Find the determinants of each minor and multiply that by the element the determinant came from
• Add each of the cofactors of the 3 solutions together

Question 15

How do you calculate the inverse of a 2x2 matrix?

Answer 15

M^-1 = 1/det M ( d -c )
………………………( -b a )

Question 16

M x M^-1 = ???

Answer 16

M x M^-1 = I

Question 17

Which of these is correct if A and B are non singular?

• (AB)^-1 = A^-1B^-1
• (AB)^-1 = B^-1A^-1

Answer 17

(AB)^-1 = B^-1A^-1

Question 18

Transpose ( a b )
…………………( c d )

Answer 18

( a c )
( b d )

Question 19

How do you find the inverse of a 3x3 matrix?

Answer 19

• Find the determinant (det A)
• Find the matrix of minors (M)
• Using M, find the matrix of cofactors (C)
• Transpose C to get C^T
• Divide C^T by det A

Question 20

Put these equations in matrix form:

x + y + z = 0
2x - 24z 13
y - x = 3

Answer 20

( 1 1 1 ) ( x ) = ( 0 )
( 2 0 -24 ) ( y ) = ( 13 )
( -1 1 0 ) ( z ) = ( 3 )

Question 21

What is the next step in solving:

( 1 1 1 ) ( x ) = ( 0 )
( 2 0 -24 ) ( y ) = ( 13 )
( -1 1 0 ) ( z ) = ( 3 )

Answer 21

Multiply both sides by:

( 1 1 1 )^-1
( 2 0 -24 )
( -1 1 0 )

Question 22

What does it mean if a system is consistent?

Answer 22

It has either:

• One solution
• Infinite solutions

Question 23

What does it mean if a system is inconsistent?

Answer 23

There are no solutions

Question 24

What does it mean if an equation corresponding to a set of linear equations is non singular and has an inverse?

Answer 24

• The system is consistent
• There is a unique solution

Question 25

What does it mean if an equation corresponding to a set of linear equations is singular and doesn’t have an inverse?

Answer 25

The system is either:

• Consistent with infinite solutions
• Inconsistent with no solutions

Question 26

Describe the configuration of planes when a system is consistent with a unique solution

Answer 26

All 3 planes meet at a point

Question 27

Describe the configuration of planes when a system is consistent with infinite solutions

Answer 27

Either:

• A sheaf where the planes meet across a line
• Or the equations all represent the same plane

Question 28

Describe the configuration of planes when a system is inconsistent with no solutions

Answer 28

Either:

• A prism where at least 2 planes meet, but they don’t all share a common point
• Or where 2 or more of the planes are parallel