M2: Matrix and Matrix algebra

Question 1

Rectangular collection of variables or scalars contained within a set of square[] or within round brackets

Answer 1

Matrix

Question 2

CLASSIFICATION OF MATRIX:
Number of rows=number of column

Answer 2

Square matrix

Question 3

CLASSIFICATION OF MATRIX:
Square matrix with all zero value except for aij where i=j

1 0 0
0 5 0
0 0 9

Answer 3

Diagonal Matrix

Question 4

Diagonal matrix with all non-zero entry equal to one
1 0 0
0 1 0
0 0 1

Answer 4

Identity matrix

Question 5

A diagonal matrix with all non-zero entry equal to some other constant
7 0 0
0 7 0
0 0 7

Answer 5

scaler matrix

Question 6

MATRIX MODIFICATION:
Multiply a row by a non zero constant

Answer 6

Scale operation

Question 7

MATRIX MODIFICATION:
Use the other row to modify the other. Example: R1=R1-2R2

Answer 7

Pivot operation

Question 8

MATRIX MODIFICATION:
Exchange two rows of the matrix

Answer 8

Swap operation

Question 9

How to add and subtract matrices

Answer 9

1. Ensure that they have the same order (ex both are 3x3 Row x Column)
2. Add/Subtract values of the same position

Question 10

How to Multiply matrices

Answer 10

1. Note that AB != BA
2. Check if the #column of A matches the # rows in B. Otherwise, it cannot be multiplied
3. The order or size can be determined by multiplying the row of a and the column of b
4. Multiply values in first row a to every values in first column b nd add the product

Question 11

How to divide

Answer 11

1. Get the inverse of b
2. Multiply

Question 12

How to calculate determinants

Answer 12

1. Get the minor
2. Consider the cofactor sign using (-1)^(J+K)
3. Perform 3X2-> 2x2 process of getting determinant (depends on the process)
4. Multiply and add values

Question 13

___ measures how much a vector space change in a matrix transformation

Answer 13

Determinants

Question 14

The __ indicates that the the vector space was flip after matrix transformation

Answer 14

Negative determinants

Question 15

PROPERTIES OF DETERMINANTS MATRIX
1. If one row or column are all zeroes, ___

Answer 15

it has zero determinant

Question 16

PROPERTIES OF DETERMINANTS MATRIX
2. If there is an identical column/row, it has ___

Answer 16

Zero determinant

Question 17

PROPERTIES OF DETERMINANTS MATRIX
3. To solve for determinant in diagonal matrix, ___

Answer 17

Multiply the product of diagonal element

Question 18

PROPERTIES OF DETERMINANTS MATRIX
4. The determinant ___ when the values of rows and columns are transposed/interchanged

Answer 18

does not change

Question 19

PROPERTIES OF DETERMINANTS MATRIX
5. If each column is multiplied by m, then ____

Answer 19

m x determinant

Question 20

PROPERTIES OF DETERMINANTS MATRIX
6. If two columns and rows are interchanged, the ___

Answer 20

sign of determinant is changed

Question 21

PROPERTIES OF DETERMINANTS MATRIX
Determinant does not change if each element of a column/row is __ by a number k and __ to the corresponding elements of a column

Answer 21

multiplied
Added/subtracted

Question 22

PROPERTIES OF DETERMINANTS MATRIX
Addition of each 1st row of two matrix equal to the ___

Answer 22

sum of the first row at the right matrix

Question 23

Inverse of a 2X2 matrix formula

Answer 23

A^(-1)= (1/(AD-BC)) | D -B |
| -C A |

Question 24

Inverse of a 3x3 matrix fomula

Answer 24

1. Make the matrix look like this
   [Matrix a| Identity of matrix A]
2. Make the upper and lower triangle of matrix A equal to zero (convert to diagonal matrix) and apply the solution to Identity as well
   A 0 0 | D 0 E
   0 B 0 | F G 0
   0 0 C | H P R
3. Make the values of diagonal matrix equal to one and apply the solution to the identity

1 0 0 | D/A 0 E/A
0 1 0 | F/B G/B 0
0 0 1 | H/C P/C R/C

4. Confirm that is is true by multiplying the AxA^(-1). If the value the an identity matrix, then it is the inverse

Question 25

Transpose of a Matrix or A^(T)

Answer 25

A=
A 1
B 1
C 1

A^(T)=
A B C
1 1 1

Question 25

Transpose of a Matrix or A^(T)

Answer 25

A=
A 1
B 1
C 1

A^(T)=
A B C
1 1 1

Question 26

How to get Eigenvalues

Answer 26

1. Use A- Lambda (Identity) then get characteristic eq by using the rules for getting determinant
2. Get Eigenvalues by factoring the eq
3. Substitute the first Eigenvalue such that A-Value1(Identity)
4. Conduct Row echelon form to get the first row values only
5. Convert matrix to equation
6. Get x1 and X2 via guessing the first value and substituting to the other
7. Repeat steps 3 onwards until all eigenvalues are substituted