Linear: 2.1 - 2.2 True/False
Each elementary matrix is invertible
True - p 108
If A can be row reduced to the identity matrix, then A must be invertible
True - p 109 - Theorem 7
End of 2.1
End of 2.1
Each column of AB is a linear combination of the columns of B using weights from the corresponding column of A.
False - Each column in AB is a linear combination of A's columns with the entry in each B column used as weights
If A and B are 2 2 with columns a1, a2, and b1, b2, respectively, then AB = [a1b1 a2b2].
False - Matrix multiplication involves the rows of A with the columns of B, not the columns of A.
(AB)C = (AC)B
False - Matrix multiplication is not commutative (AB)C != (AC)B - p 99 - because the ORDER of the matrices being multiplied matters
If A and B are 3 x 3 and B = [b1 b2 b3], then AB = [Ab1 + Ab2 + Ab3].
False - There should not be addition signs present. The product must be a 3 x 3 matrix.
(AB)^T = A^T B^T
False - p 101 - Theorem 3 - (AB)^T = B^T A^T
If A and B are n x n and invertible, then A^-1 B^-1 is the inverse of AB.
False - p 107 - Theorem 6 - (AB)^-1 = B^-1 A^-1
A product of invertible n x n matrices is invertible, and the inverse of the product is thhe product of their inverses in the same order.
False - p 108 - The inverse is the product of their inverses in REVERSE order
If A is invertible, then elementary row operations that reduce A to the identity In, the identity matrix, also reduce A^-1 to In
False - p 109 - Theorem 7 - An n x n matrix A is invertible only if A is row equivalent to In, and in this case, any sequence of elementary row operations that reduces A to In also transforms In into A^-1
In order for a matrix B to be the inverse of A, both equations AB = I and BA = I must be true.
True - by the definition of invertible matrix - p 105
AB + AC = A(B + C)
True - by the left distributive property. Matrix multiplication distributes over addition.
The second row of AB is the second row of A multiplied on the right by B.
True - by the row-column rule
A^T + B^T = (A + B)^ T
True - p 101 - Theorem 3
The transpose of a product of matrices equals the product of their transposes in the same order.
True - p 101 - Theorem 3
The transpose of a sum of matrices equals the sum of their transposes
True - p 101 - Theorem 3
If A is invertible, then the inverse of A^-1 is A itself
True - p 105
If A = [ a b] and ab - cd != 0, then A is invertible [ c d]
True - p 105 - Theorem 4
If A = [ a b] and ad = bc, then A is not invertible [ c d]
True - p 105 - Theorem 4
If A is an invertible n n matrix, then the equation Ax = b is consistent for each b in Rn.
True - p 106 - Theorem 5
