Matrices: Structure, Operations, Transformations, and Applications

All flashcards

What is a matrix?

A rectangular array of entries arranged in horizontal rows and vertical columns.

What does the size $m\times n$ mean?

The matrix has $m$ rows and $n$ columns, in that order.

What is a square matrix?

A matrix with the same number of rows and columns.

How is the entry in row $i$, column $j$ denoted?

It is commonly denoted $a_{ij}$ or $a_{i,j}$; $i$ identifies the row and $j$ identifies the column.

What are row and column vectors?

A row vector has exactly one row; a column vector has exactly one column.

Quiz questions

1. For matrices $A$ and $B$, when is the sum $A+B$ defined?Answer: When $A$ and $B$ have the same dimensions.
2. If $A$ is $m\times n$ and $B$ is $n\times p$, what is the size of $AB$?Answer: $m\times p$
3. If $A$ maps vectors from $\mathbb{R}^n$ to $\mathbb{R}^m$ and $B$ maps from $\mathbb{R}^m$ to $\mathbb{R}^k$, which product represents applying $A$ first and then $B$?Answer: $BA$
4. When is a square matrix invertible?Answer: When its determinant is nonzero.
5. What condition defines an eigenvector $v$ of a matrix $A$ with eigenvalue $\lambda$?Answer: $Av=\lambda v$, with $v$ nonzero.