Y1, C6 - Matrices

Question 1

How many rows and columns does a 3 x 2 matrix have

Answer 1

3 rows
2 columns

Question 2

How to add or subtract matrices

Answer 2

Add / subtract the corresponding matrix

Question 3

What is a zero matrix

Answer 3

All elements are 0

Question 3

What is an identity matrix

Answer 3

Square matrix 0s filled with 1s in the leading diagonal

Question 4

What are the only matrices that can be raised to a power
A

Answer 4

Square matrices

Question 5

When can you multiply two matrices

Answer 5

When the dimensions agree
The columns of matrix A must equal the rows of matrix B e.g.
(3 x 4) * (4 x 5)

Question 6

Are matrices commutative or noncommutative and what does this mean

Answer 6

Noncommutative
The order in which you multiply matrices changes the output

Question 6

What happens when you multiply by an identity matrix (A * I) OR (I * A)

Answer 6

The result will equal A

Question 7

Matrix multiplication is associative, what does this mean

Answer 7

A(BC) = (AB)C
it doesn’t matter what order you multiply in

Question 8

What does the determinant of a matrix tell you

Answer 8

If it has an inverse

Question 9

How do you know if a matrix has an inverse

Answer 9

If the determinant is NOT 0

Question 10

What is a singular matrix

Answer 10

A matrix with no inverse, determinant = 0

Question 11

What is a non-singular matrix

Answer 11

A matrix with an inverse, determinant greater or less than 0

Question 12

Formula for the determinant of a 2 x 2 matrix

Answer 12

(ad) - (bc)

Question 13

Formula for determinant of a 3 x 3 matrix

Answer 13

a(ei - fh) - b(di - fg) + c(dh - eg)
minor of A - minor of B + minor of C

Question 14

What is the minor of a 3 x 3 matrix

Answer 14

The determinant of the 2 x 2 matrix left over after the rows and columns of the element you are working out for the minor for have been crossed out

Question 15

What is M x M^-1

Answer 15

I

Question 16

What does it mean if a matrix is self inverse

Answer 16

M = M^-1
M * M = I
M^2 = I

Question 17

How do you inverse a 2 x 2 matrix

Answer 17

A^-1 = 1 / (det(A)) (d , -b, -c, a)
(1 / determinant of A) * matrix with a and b swapped, b and c negated)

Question 18

BAB = I, prove A = B^-1 * B^-1

Answer 18

BAB = I
B^-1 BAB = B^-1 I
IAB = B^-1
AB = B^-1
A = B^-1 B^-1

Question 19

Prove that (PQ)^-1 = Q^-1 * P^-1

Answer 19

(PQ)(PQ)^-1 = I
Q(PQ)^-1 = P^-1
(PQ)^-1 = Q-1 * P^-1

Question 20

What is the transpose of a matrix (A^T)

Answer 20

The rows and columns are interchanged
m x n becomes n x m

Question 21

What is a matrix of cofactors (C)

Answer 21

A matrix which follows the pattern of +, -, +, -

Question 22

How to find the inverse of a 3 x 3 matrix

Answer 22

1) Find det(A)
2) Form a matrix of minors
3) Form a matrix of cofactors (negate every other element)
4) A^-1 = (1 / det(A)) * C^T
Inverse = (1 / determinant) * Transpose of matrix of cofactors

Question 23

How can you use a matrix to solve simultaneous equations

Answer 23

If A(x, y, z) = v then (x, y, z) = A^-1 * v

Question 24

What does it mean if a system of equations is inconsistent

Answer 24

There are no solutions

Question 25

What is a sheaf

Answer 25

When 3 planes intersect along a line

Question 26

What is a prism in matrices

Answer 26

When planes intersect in pairs but don’t all intersect at any one point

Question 27

What does it mean if a system of equations is consistent

Answer 27

There is at least one solution of all three equations

Question 28

How would you know if 3 simultaneous equations are consistent or inconsistent, if there have one, zero, or infinite solutions and if they: meet at a point, a plane, a sheaf, parallel, or create a prism

Answer 28

No solutions, parallel or prism, use 2 of the equations to eliminate a variable and if remaining equations are multiples it is parallel
If infinite solutions, sheaf or same planes, if all planes are multiples they are a plane
One solution, all planes meet at one point