Matrices

Question 1

What is a diagonal matrix?

Answer 1

A square matrix where all elements not on the main diagonal are 0

Question 2

What is a lower triangular matrix?

Answer 2

A square matrix that only has nonzero values on the main diagonal & below

Question 3

What is an upper triangular matrix?

Answer 3

A square matrix that only has nonzero values on the main diagonal & above

Question 4

What is a boolean or binary matrix?

Answer 4

A matrix that contains only 0s & 1s

Question 5

What is a right or row stochastic matrix?

Answer 5

A square matrix where each value is between 0 & 1, inclusive, & the sum of each row is 1

Question 6

What is a left or column stochastic matrix?

Answer 6

A square matrix where each value is between 0 & 1, inclusive, & the sum of each column is 1

Question 7

How do you add matrices?

Answer 7

add elements in the same position of both matrices. The result will be a matrix of the same size.

Question 8

What is the precondition of adding matrices?

Answer 8

They must be of the same order.

Question 9

What is the order of a matrix?

Answer 9

The dimensions (rows x columns)

Question 10

What is the precondition to find a dot product?

Answer 10

They must be vectors of the same length, but opposite orientation (one row vector & one column vector)

Question 11

How do you find a dot product?

Answer 11

Multiply the elements of a row vector with elements with the same index of from a column vector, then add the results. The result is a scaler.

Question 12

What is the precondition to find the product of 2 matrices?

Answer 12

The number of columns in the first matrix must equal the number of rows in the second. If you write the orders next to each other, the inner numbers must match. m x p, p x n

Question 13

What is the order of the product of 2 matrices?

Answer 13

The resulting matrix has the same number of rows as the first matrix & the same number of columns as the second. If you write the orders next to each other, the outer numbers represent the order of the result. m x p * p x n = m x n

Question 14

Practice finding the product of 2 matrices.

Answer 14

Did you do well? Do it on your calculator to confirm

Question 15

What is a vector?

Answer 15

A matrix with only one row or column.

Question 16

What is a scaler?

Answer 16

A regular number or 1x1 matrix (same thing)

Question 17

What is a matrix containing only 0s called?

Answer 17

A null matrix

Question 18

What is the main diagonal?

Answer 18

All the elements of a matrix whose row & column indices are the same, ie the diagonal starting from the top left corner. Only in the case of a square will it reach the bottom right corner.

Question 19

What is the identity matrix?

Answer 19

a square matrix with only 1 on the main diagonal, & only 0 elsewhere. Denoted I[4], where 4 is the order.

Question 20

Is a node adjacent to itself in a graph’s adjacency matrix?

Answer 20

Yes

Question 21

When are matrices equal?

Answer 21

When they are the same order & each element is the same

Question 22

What happens when the preconditions for matrix operations are not met?

Answer 22

The result is undefined.

Question 23

What is the precondition for squaring a matrix?

Answer 23

The order of the matrix must be square.

Question 24

What is the transpose of a matrix?

Answer 24

It’s when you swap the rows with the columns or mirror it across the main diagonal. Notated A^t or A^T (superscript). The order of the transpose will be the opposite of the original. The main diagonal will be unchanged.

Question 25

What is a symmetric matrix?

Answer 25

a matrix that is equal to its transposition or a matrix that is symmetrical along its main diagonal.

Question 26

What is the precondition of finding the determinant?

Answer 26

The matrix must be square.

Question 27

What does |A| mean?

Answer 27

The determinant of A

Question 28

Practice finding the determinant of a 4x4 matrix using Laplace Expansion & the 2x2 method.

Answer 28

Do it on your calculator to check.

Question 29

Practice finding the determinant of a 4x4 matrix using Laplace Expansion & Sarrus’s Rule for 3x3.

Answer 29

Check with your calculator

Question 30

What are the downward diagonals?

Answer 30

The diagonals parallel to the main diagonal which are the same length in a wide matrix

Question 31

What are the upward diagonals?

Answer 31

The diagonal starting in the bottom left corner & those parallel to it which are the same length in a wide matix, especially an augmented matrix

Question 32

How do you find the determinant of a 2 x 2 matrix?

Answer 32

The product of the main diagonal minus the product of the others

Question 33

How do you find the determinant using the Sarrus Method?

Answer 33

Write the first 2 columns again after the last column in the same order, then find the product of each of the downward & upward diagonals. Subtract the sum of the product of the the upward diagonals from the downward.

Question 34

What is the minor of an element of a matrix?

Answer 34

The determinant of the matrix created by removing the row & column on which that element resides

Question 35

What is the notation for the minor?

Answer 35

M[ij]

Question 36

What is the notation for the cofactor?

Answer 36

C[ij]

Question 37

What is the formula for the cofactor?

Answer 37

C[ij] = (-1)^(i+j)*M[ij]

Question 38

How do you find the determinant using laplace expansion?

Answer 38

the sum of an element times its cofactor for an entire row or column

Question 39

What is a non-singular matrix?

Answer 39

a matrix whose determinant is not 0.