FM - Core Pure 1 - 6) Matrices

Question 1

Definition of a square matrix

Answer 1

Matrix where the numbers of rows and cloumns are the same

Question 2

Definition of a zero matrix

Answer 2

Matrix where all the numbers are 0

Question 3

Definition and notation of an identity matrix

Answer 3

• Square matrix where the numbers in the leading diagonal (TL to BR) are 1 and the rest are 0
• I_k where k is the number of rows (or columns)

Question 4

Result of multiplying a matrix by its inverse matrix

Answer 4

MM^-1 = I

Question 5

Information that the rows and colums of matrices give when multiplying them

Answer 5

R×C × R×C ↓

• Inside numbers show if the two can be multiplied
• Outside numbers show the dimensions of the resulting multiplied matrix

Question 6

Definition of a singular and non-singular matrix

Answer 6

• Singular → determinant = 0
• Non-singular → determinant ≠ 0

Question 7

Determinant of a 2×2 matrix

Answer 7

ad - bc

Question 8

Inverse of a 2×2 matrix

Answer 8

1/det(M) |^d_-c^-b_a|

Question 9

Determinant of a 3×3 matrix

Answer 9

2×2 determinants of → a|^e_h^f_·i| - b|^d_g^f_·i| + c|^d_e^g_h|

Question 10

Method to find the inverse of a 3×3 matrix

Answer 10

1. Find the determinant
2. Form the matrix of minors (top 3 of 9 already done)
3. Form the matrix of cofactors (inverse sign for middles of each edge)
4. Transpose it (rows become columns)
5. 1/determinant |transposed matrix|

Question 11

Manipulate (AB)^-1 (non-singlular matrices)

Answer 11

B^-1A^-1

Question 12

Rearrange (matrix) A(xyz) = v (xyz is vertical)

Answer 12

(xyz) = A^-1v

Question 13

Meanings of a consistent and inconsistent system of linear equations

Answer 13

• Has one set of values that satisfy all the equations simultaneously
• Otherwise its inconsistent

Question 14

Method to find the geometric configuration of the corresponding matrix what each means

Answer 14

1. Non-singular → consistent, one unique point, planes meet at a single point
2. LHS and RHS of each equation are multiples of all the others → consistent, infinite many solutions, all three equations represent the same plane
3. LHS (but not RHS) of two or more equations are multiples of each other → inconsistent, no solutions, two or more of the planes are parallel and non-identical
4. When one variable is eliminated, there are solutions to the new equations → consistent, infinite many solutions, sheaf
5. ↑ But has no solutions→ inconsistent, no solutions, prism