1.3 Matrix Operations Flashcards
What is a matrix?
A matrix is a rectangular array of numbers. The numbers in the array are called the entries in the matrix.
How do we describe the size of a matrix?
Always rows first, then columns.
rows x columns.
What is a matrix with only one row called?
A row vector (or row matrix).
What is a matrix with only one column called?
A column vector (or column matrix).
A matrix with n rows and n columns we call…
A square matrix of order n.
In a square matrix, A, the entries a11, a22, … , ann are collectively known as…
The main diagonal of A.
What is the definition of two matrices being equal?
Two matrices are equal if they are of the same size and their corresponding elements are equal.
(A)ij = (B)ij for all values of i and j.
How do we define matrix addition?
The sum A + B is the matrix obtained by adding the entries of B to the corresponding entries of A.
So, if A and B are the same size, and A = [aij] and B = [bij], then…
(A + B)ij = (A)ij + (B)ij = aij + bij
for all values of i and j
How do we define matrix subtraction?
The difference A - B is the matrix obtained by subtracting the entries of B from the corresponding entries of A.
So, if A and B are the same size, and A = [aij] and B = [bij], then…
(A - B)ij = (A)ij - (B)ij = aij - bij
for all values of i and j
How do we define scalar multiplication?
If A is any matrix and c is any scalar, then the product cA is the matrix obtained by multiplying each entry of A by c. The resulting matrix said to be a scalar multiple of A.
So, if A = [aij], then (cA)ij = c(A)ij = caij
It is common practice to denote the scalar multiple (-1)B by…
-B
What is the result of adding (or subtracting) two matrices of different sizes?
This operation is undefined.
What is the size of the product of A and B, where A is m x r and B is r x n?
Size of AB is m x n.
What is the requirement on the operands for matrix multiplication to be defined?
For AB to be defined, the number of columns on A must equal the number of rows on B.
What is the size of the product of A and B?
AB, if defined, will have the number of rows of A x the number of columns of B.
What is an augmented matrix?
It’s an abbreviation of a system of linear equations.
The entry that occurs in row i and column j of a matrix A is usually denoted by…
aij
How do we define matrix multiplication?
Given matrices A (size m x r) and B (size r x n), we find the entry ij of AB by taking the i-th row of A and the j-th column of B, multiply the corresponding entries of the row and column together, then sum the resulting products.
How do we compactly denote the generalised contents of a matrix A?
[aij]
So, we can say A = [aij] and (A)ij = aij
How do we define the transpose of a matrix?
The transpose results by interchanging the rows and columns. Formally:
(At)ji = (A)ij
What do we observe about the transpose of a square matrix?
It can be obtained by ‘reflecting’ the entries about the main diagonal.
How do we define the trace of a matrix?
Denoted by tr(A), is the sum of the entries along the main diagonal. Undefined if the matrix is not square.
Formally:
if A = [aij]nxn
tr(A) = a11 + a22 + a33 + … + ann
What is tr(A)?
The trace of A - the sum of the entries along the main diagonal.
What do we call the sum of the entries along the main diagonal of a square matrix?
The trace.
