Chapter 2 Matrix Algebra
Define diagonal matrix and zero matrix
A diagonal matrix is a square matrix who's nondiagonal entries are all zero. The zero matrix is the matrix whose entries are all zeros
If AB=I what can we say about A and B?
A is the inverse of B and B is the inverse of A (pg. 114)
What is an elementary matrix?
A matrix that is obtained by performing a single elementary row operation on an identity matrix.
What is a singular matrix? What is a nonsingular matrix?
A singular matrix is a matrix that is NOT invertible. A nonsingular matrix is invertible (pg. 105)
Define what it means for a matrix to be invertible
An nxn matrix is invertible if there exists a matrix A⁻¹ such that AA⁻¹=In AND A⁻¹A=In (pg. 105)
What are partitions? What is a partitioned (AKA block) matrix?
Partitions are horizontal and vertical lines that split matrices up into smaller rectangular arrays. A partitioned matrix is one that has been split up by partitions and written in a more compact form. See example 1 (pg. 119).
Define transpose
pg. 100
Given the matrix ABC, find its transpose.
pg. 101
Theorem involving transposes
pg. 101
Redo problem 25 from 2.1
pg. 103
Formula for finding the inverse of an invertible 2x2 matrix
pg. 105
Explain the following
pg. 108
Given the matrix ABC, find its inverse
pg. 108
Explain the following statement
pg. 109
Theorem 7
pg. 109
Algorithm for finding the inverse of a matrix.
pg. 110
What does it mean if a linear transformation is invertible?
pg. 115
Theorem 1: 6 matrix properties
pg. 95
Given A and B, find ABij (assuming multiplication is defined)
ABij is found by multiplying the ith row of A with the jth column of B
A 2x2 matrix, A, is invertible iff ...
Det(A)≠0
Explain how to multiply partitioned matrices.
For a matrix product AB, the partitions of the columns of A must correspond to the partitions of the rows of B. If this is the case, then we say that A and B are conformable for block multiplication. See example 3 (pg. 120)
What is the inverse of the transpose of a matrix? Prove it. (Theorem 6c)
Given A and its transpose, the inverse of the transpose is the transpose of the inverse of A.
What is the inverse of the product of two matrices? Prove it. (Theorem 6b)
Given AB, the inverse is B⁻¹A⁻¹. (pg. 107)
Explain aij notation. What does aj denote?
Given an mxn matrix, aij is the term in the ith row and jth column. When we write aj, we denote the jth column of A.
What does it mean to transpose?
Given an mxn matrix, it's transpose is an nxm matrix where the rows and columns have been switched
State theorem five and prove it.
It tells us that if A is invertible, then the equation Ax=b has a unique solution A⁻¹b (pg. 106)
Explain how to find the determinant of a 2x2 matrix. What does the determinant tell us?
The determinant tells us whether or not the matrix is invertible. If the Det(A)=0, then A is singular (not invertible). If Det(A)≠0 then A in nonsingular
Given a matrix, A, and its inverse A⁻¹, what is the inverse of A⁻¹? Prove it. (Theorem 6a)
The inverse of the inverse is just A.
Discuss matrix multiplication in terms of transformation composition
The linear transformation defined by T(x) = (AB)x is the same as the linear transformation obtained my first mapping x → Bx, followed by mapping Bx→ A(Bx).
Invertible matrix theorem. This only applies to what type of matrices? Prove it.`
This only applies to SQUARE matrices (pg. 114)
Define matrix addition and scalar multiplication
To add/subtract matrices, they must be the same size. The sum is the matrix whose entries are the sum of the corresponding matrices. To multiply a matrix by a scalar, distribute the scalar to every entry in the matrix
Define matrix multiplication
To multiply matrices, the number of columns in the first must match the number of rows in the second. The resulting matrix will have the number of rows as the first and the number of columns of the second. The jth column of AB is A times the jth column of B. (pg. 97). Each column of AB is a linear combination of the columns of A using weights from the corresponding column of B.
