Chapter 2- Matrix Algebra Flashcards
Properties of Matrix Multiplication (Theorem 2)
1) A(BC) = (AB)C
2) A(B + C) = AB + AC
3) (B + C)A = BA + CA
4) r(AB) = (rA)B = A(rB) r is scalar
5) ImA = A = AIn
6) A^k = AAA … *A (k times)
Warnings about multiplying matrices
1) In general, AB =/= BA
2) Cancellation laws do NOT hold for matrices. That is, if AB = AC, then in general, B =/= C.
3) If AB = 0, that does NOT imply A = 0 or B = 0
Definition of transposing a matrix A
A is an mxn matrix, then the transpose of A, denoted by A^T, whose switching the rows and columns to an nxm matrix
What are the properties of transposing a matrix?
Let A and B denote matrices whose sizes are appropriate for the following sums and products.
1) (A^T)^T = A
2) (A + B)^T = A^T + B^T
3) for any scalar r, (rA)^T = rA^T
4) (AB)^T = B^T*A^T
The transpose of a product of matrices equals the product of their transposes in the reverse order
If A and B are matrices, how do we know if AB is defined?
AB is defined iff the number of columns for A is equal to the number of rows for B
How to compute AB separately using its components
Each column of AB is a linear combination of the columns of A using weights from the corresponding column of B (B has p columns).
AB = A[b1 b2 b3 … bp]
Let A and B be matrices. If A and B commute with one another…
… then AB = BA. Also, they are both square matrices.
If a matrix A is 5x3 and the product AB is 5x7, what is the size of B?
3x7
True or false. Why?
If A and B are 2x2 matrices with columns a1, a2, and b1, b2, respectively, then AB = [a1b1 a2b2]
False
AB = A[b1 b2] = [Ab1 Ab2]
True or false. Why?
Each column of AB is a linear combination of the columns of A using weights from the corresponding column of B.
True
AB = [Ab1 Ab2 … Abp]
True or false. Why?
The transpose of a product of matrices equals the product of their transposes in the same order.
False
(AB)^T = B^T*A^T
Says the property of transposing matrices
True or false. Why?
If A is an nxn matrix,
then (A^2)^T = (A^T)^2
True
(A^2)^T = (AA)^T = A^TA^T
= (A^T)^2
Suppose the third column of B is the sum of the first two columns. What can be said about the third column of AB? Why?
Third column of AB is the sum of the first two columns of AB
Third column AB = Ab3 = A(b1 + b2)
= Ab1 + Ab2
By a property of matrix-vector multiplication
Suppose the first two columns, b1 and b2, of B are equal, what can we say about AB. Why?
Those columns will be equal to each other
AB = [Ab1 Ab2 … ]
If b1 and b2 are equal, the first two columns of AB are interchangeable.
Suppose the last column of AB is entirely zeros but B itself has no column of zeros. What can be said about the columns of A? Why?
Columns of AB are linearly dependent. Figure out why
True or false. Why?
If A and B are nxn and invertible, then A^(-1)B^(-1) is the inverse of AB
False.
The product of nxn invertible matrices is invertible, and the inverse is the product of their inverses in the REVERSE order.
States theorem 6
If A is a 2x2 matrix, what is the determinant of A and what can that tell us?
Det(A) = ad - bc
Det(A) =/= 0, then A is invertible
Det(A) = 0, then A is NOT invertible
What is a singular matrix?
What is a nonsingular matrix?
Singular matrix: matrix that is NOT invertible
Nonsingular matrix: matrix that is invertible
Theorem 5?
If A is an invertible matrix, what can we say about Ax = b?
If A is an invertible nxn matrix, then for each b in R^n, the equation
Ax = b has the unique solution
x = A^(-1)b
Theorem 6
Facts about invertible matrices
1) (A^(-1))^(-1) = A
2) (AB)^(-1) = B^(-1)A^(-1)
The inverse of AB is the product of the inverses of A and B in the reverse order.
3) (A^(T))^(-1) = (A^(-1))^(T)
Theorem 7
Best way to visualize an invertible matrix. So pay attention!
An nxn matrix A is invertible iff A is row equivalent to I, and in this case, and sequence of elementary row operations that reduces A to I also transforms I to A^(-1)
Algorithm for finding A^(-1)
If we place A and I side-by-side to form an augmented matrix
[A I], then row operations to transform A to I will also transform I to A^(-1)
True or false. Why?
If A is invertible, then the inverse of
A^(-1) is A itself
True
What is the invertible matrix theorem? (IMT)
Let A be a square nxn matrix. These statements are either all true or all false.
1) A is an invertible matrix
2) A is row equivalent to I (n pivot positions)
3) Ax = b has only one solution for each b in R^n (existence and uniqueness)
4) Columns of A span R^n
5) A^T is invertible
True or false. Why?
If A is invertible, then the columns of A are linearly dependent.
False
Remember the algorithm for finding A-inverse. Must reduce A to rref with pivots in every row and column
What is an upper triangular matrix?
Lower triangular matrix?
Upper: Matrix whose entries below the main diagonal are all zero’s
Lower: entries above main diagonal are all zero’s
Can a square matrix with two identical columns be invertible? Explain
No because those columns are linearly dependent, violation IMT
What is an upper triangular matrix?
Lower triangular matrix?
Upper: Matrix whose entries below the main diagonal are all zero’s
Lower: entries above main diagonal are all zero’s
Can a square matrix with two identical columns be invertible? Explain
No because those columns are linearly dependent, violation IMT
