True/False Chapter 2 Flashcards
If A and B are 2x2 with columns a1, a2 and b1, b2, respectively, then AB=[a1 b1 a2 b2]
false
Each column of AB is a linear combination of the columns of B using weights from the corresponding columns of A
false : the roles of A and B should be reversed in the second half statement
AB + AC = A(B+C)
true
A^T + B^T = (A+B)^T
true
The transpose of a product of matrices equals the product of their transposes in the same order
false : the phrase ‘in the same order’ should be ‘in the reverse order’
If A and B are 3x3 and B=[b1 b2 b3], then AB=[Ab1+Ab2+ab3]
false : AB must be a 3x3 matrix, but the formula for AB implies that it is 3x1. The plus sign should just be spaces (between columns)
The second row of AB is the second row of A multiplied on the right by B
true
(AB)C = (AC)B
false : the left-to-right order of B and C cannot be changed, in general
(AB)^T = A^T B^T
false
The transpose of a sum of matrices equals the sum of their transposes
true
A product of invertible nxn matrices is invertible, and the inverse of the product is the product of their inverses in the same order
false : the product matrix is invertible, but the product of inverses should be in the reverse order
If A is invertible, then the inverse of A^-1 is A itself
true
If A = a b and ad=bc then A is not invertible
c d
true
If A can be row reduced to the identity matrix, then A must be invertible
true
If A is invertible, then elementary row operations that reduce A to the identity In also reduce A^-1 to In
false
In order for a matrix B to be the inverse of A, both equations AB = I and BA = I must be true
true
If A and B are nxn and invertible, then A^-1B^-1 is the inverse of AB
false
If A = a b and ab-cd /= 0, then A is invertible
c d
false : it’s ad-bc /=0
If A is an invertible nxn matrix, then the equation Ax=b is consistent for each b in Rn
true
Each elementary matrix is invertible
true
If the equation Ax=0 has only the trivial solution, then A is row equivalent to the nxn identity matrix
true : by IMT, if statement d is true, then so is b
If the columns of A span Rn, then the columns are linearly independent
true : by IMT
If A is an nxn matrix, then the equation Ax=b has at least one solution for each b in Rn
false : only true for invertible matrices
If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot positions
true
If A^T is not invertible, then A is not invertible
true
If there is an nxn matrix D such that AD = I, then there is also an nxn matrix C such that CA = I
true
If the columns of A are linearly independent, then the columns of A span Rn
true
If the equation Ax=b has at least one solution for each b in Rn, then the solution is unique for each b
true
If the linear transformation (x)–>Ax maps Rn into Rn, then A has n pivot positions
false : the first part of the statement is not part i of the IMT. In fact, if A is any nxn matrix, the linear transformation x–>Ax maps Rn into Rn, yet not every such matrix has n pivot positions
If there is a b(vector) in Rn such that the equation Ax=b is inconsistent, then the transformation x–>Ax is not one-to-one
true
