Matrices Flashcards
What is a diagonal matrix?
A square matrix where all elements not on the main diagonal are 0
What is a lower triangular matrix?
A square matrix that only has nonzero values on the main diagonal & below
What is an upper triangular matrix?
A square matrix that only has nonzero values on the main diagonal & above
What is a boolean or binary matrix?
A matrix that contains only 0s & 1s
What is a right or row stochastic matrix?
A square matrix where each value is between 0 & 1, inclusive, & the sum of each row is 1
What is a left or column stochastic matrix?
A square matrix where each value is between 0 & 1, inclusive, & the sum of each column is 1
How do you add matrices?
add elements in the same position of both matrices. The result will be a matrix of the same size.
What is the precondition of adding matrices?
They must be of the same order.
What is the order of a matrix?
The dimensions (rows x columns)
What is the precondition to find a dot product?
They must be vectors of the same length, but opposite orientation (one row vector & one column vector)
How do you find a dot product?
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.
What is the precondition to find the product of 2 matrices?
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
What is the order of the product of 2 matrices?
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
Practice finding the product of 2 matrices.
Did you do well? Do it on your calculator to confirm
What is a vector?
A matrix with only one row or column.
What is a scaler?
A regular number or 1x1 matrix (same thing)
What is a matrix containing only 0s called?
A null matrix
What is the main diagonal?
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.
What is the identity matrix?
a square matrix with only 1 on the main diagonal, & only 0 elsewhere. Denoted I[4], where 4 is the order.
Is a node adjacent to itself in a graph’s adjacency matrix?
Yes
When are matrices equal?
When they are the same order & each element is the same
What happens when the preconditions for matrix operations are not met?
The result is undefined.
What is the precondition for squaring a matrix?
The order of the matrix must be square.
What is the transpose of a matrix?
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.
What is a symmetric matrix?
a matrix that is equal to its transposition or a matrix that is symmetrical along its main diagonal.
What is the precondition of finding the determinant?
The matrix must be square.
What does |A| mean?
The determinant of A
Practice finding the determinant of a 4x4 matrix using Laplace Expansion & the 2x2 method.
Do it on your calculator to check.
Practice finding the determinant of a 4x4 matrix using Laplace Expansion & Sarrus’s Rule for 3x3.
Check with your calculator
What are the downward diagonals?
The diagonals parallel to the main diagonal which are the same length in a wide matrix
What are the upward diagonals?
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
How do you find the determinant of a 2 x 2 matrix?
The product of the main diagonal minus the product of the others
How do you find the determinant using the Sarrus Method?
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.
What is the minor of an element of a matrix?
The determinant of the matrix created by removing the row & column on which that element resides
What is the notation for the minor?
M[ij]
What is the notation for the cofactor?
C[ij]
What is the formula for the cofactor?
C[ij] = (-1)^(i+j)*M[ij]
How do you find the determinant using laplace expansion?
the sum of an element times its cofactor for an entire row or column
What is a non-singular matrix?
a matrix whose determinant is not 0.
