M2: Matrix and Matrix algebra Flashcards
Rectangular collection of variables or scalars contained within a set of square[] or within round brackets
Matrix
CLASSIFICATION OF MATRIX:
Number of rows=number of column
Square matrix
CLASSIFICATION OF MATRIX:
Square matrix with all zero value except for aij where i=j
1 0 0
0 5 0
0 0 9
Diagonal Matrix
Diagonal matrix with all non-zero entry equal to one
1 0 0
0 1 0
0 0 1
Identity matrix
A diagonal matrix with all non-zero entry equal to some other constant
7 0 0
0 7 0
0 0 7
scaler matrix
MATRIX MODIFICATION:
Multiply a row by a non zero constant
Scale operation
MATRIX MODIFICATION:
Use the other row to modify the other. Example: R1=R1-2R2
Pivot operation
MATRIX MODIFICATION:
Exchange two rows of the matrix
Swap operation
How to add and subtract matrices
- Ensure that they have the same order (ex both are 3x3 Row x Column)
- Add/Subtract values of the same position
How to Multiply matrices
- Note that AB != BA
- Check if the #column of A matches the # rows in B. Otherwise, it cannot be multiplied
- The order or size can be determined by multiplying the row of a and the column of b
- Multiply values in first row a to every values in first column b nd add the product
How to divide
- Get the inverse of b
- Multiply
How to calculate determinants
- Get the minor
- Consider the cofactor sign using (-1)^(J+K)
- Perform 3X2-> 2x2 process of getting determinant (depends on the process)
- Multiply and add values
___ measures how much a vector space change in a matrix transformation
Determinants
The __ indicates that the the vector space was flip after matrix transformation
Negative determinants
PROPERTIES OF DETERMINANTS MATRIX
1. If one row or column are all zeroes, ___
it has zero determinant
PROPERTIES OF DETERMINANTS MATRIX
2. If there is an identical column/row, it has ___
Zero determinant
PROPERTIES OF DETERMINANTS MATRIX
3. To solve for determinant in diagonal matrix, ___
Multiply the product of diagonal element
PROPERTIES OF DETERMINANTS MATRIX
4. The determinant ___ when the values of rows and columns are transposed/interchanged
does not change
PROPERTIES OF DETERMINANTS MATRIX
5. If each column is multiplied by m, then ____
m x determinant
PROPERTIES OF DETERMINANTS MATRIX
6. If two columns and rows are interchanged, the ___
sign of determinant is changed
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
multiplied
Added/subtracted
PROPERTIES OF DETERMINANTS MATRIX
Addition of each 1st row of two matrix equal to the ___
sum of the first row at the right matrix
Inverse of a 2X2 matrix formula
A^(-1)= (1/(AD-BC)) | D -B |
| -C A |
Inverse of a 3x3 matrix fomula
- Make the matrix look like this
[Matrix a| Identity of matrix A] - 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 - 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
- Confirm that is is true by multiplying the AxA^(-1). If the value the an identity matrix, then it is the inverse
Transpose of a Matrix or A^(T)
A=
A 1
B 1
C 1
A^(T)=
A B C
1 1 1
Transpose of a Matrix or A^(T)
A=
A 1
B 1
C 1
A^(T)=
A B C
1 1 1
How to get Eigenvalues
- Use A- Lambda (Identity) then get characteristic eq by using the rules for getting determinant
- Get Eigenvalues by factoring the eq
- Substitute the first Eigenvalue such that A-Value1(Identity)
- Conduct Row echelon form to get the first row values only
- Convert matrix to equation
- Get x1 and X2 via guessing the first value and substituting to the other
- Repeat steps 3 onwards until all eigenvalues are substituted
