MATH 240 SECTIONS 2.1-2.3, 2.6, 3.1-
3.3, 4.1-4.3, 4.5 QUESTIONS AND
CORRECT ANSWERS
2.1 Theorem 3 ✅✅ANSW-Let A and B denote matrices whose sizes are appropriate for the
following sums and products.
a. (A^T)^T = A
b. (A + B)^T = A^T + B^T
c. For any scalar r, (rA)^T = rA^T
d. (AB)^T = B^T * A^T
The transpose of a product of matrices equals the product of their transposes in reverse order.
2.2 Theorem 5 ✅✅ANSW-If A is an invertible n x n matrix then for each b in R^n, the equation Ax
= b has the unique solution x = A^-1 * b
2.2 Theorem 6 ✅✅ANSW-a. If A is an invertible matrix, then A^-1 is invertible and the inverse of
it is A.
b. If A and B are n x n matrices, then so is AB, and the inverse of AB is the product of the inverses of
A and B in the reverse order. That is:
(AB)^-1 = B^-1 * A^-1
c. If A is an invertible matrix, then so is A^T, and the inverse of A^T is the transpose of A^-1. That is:
(A^T)^-1 = (A^-1)^T
2.2 Theorem 7 ✅✅ANSW-An n x n matrix A is invertible if and 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
2.3 Theorem 8 ✅✅ANSW-Let A be a square n x n matrix. Then the following statements are
equivalent. That is, for a given A, the statements are either all true or all false.
, a. A is an invertible matrix.
b. A is row equivalent to the n x n identity matrix.
c. A has n pivot positions.
d. The equation Ax = 0 has only the trivial solution.
e. The columns of A for a linearly independent set.
f. The linear transformation x to Ax maps R^n onto R^n
g. The equation Ax = b has at least on solution for each b in R^n
h. The columns of A span R^n
i. The linear transformation x to Ax is one-to-one
j. There is an n x n matrix C such that CA = I
k. There is an n x n matrix D such that AD = I
A^T is an invertible matrix.
2.3 Theorem 9 ✅✅ANSW-Let T : R^n to R^n be a linear transformation and let A be the standard
matrix for T. Then T is invertible if and only if A is an invertible matrix. In that case, the linear
transformation S given by S(x) = A^-1 * x is the unique function satisfying equations (1) and (2),
2.1 Theorem 1 ✅✅ANSW-Let A, B, and C be matrices of the same size and let r and s be scalars.
a. A + B = B + A
b. (A + B) + C = A + (B + C)
c. A + 0 = A
d. (r(A + B) = rA + rB
e. (r + s)A = rA + sA
f. r(sA) = (rs)A
2.1 Definition ✅✅ANSW-If A is an m x n matrix, and if B is an n x p matrix with columns b1 ... bp,
then the product AB is the m x p matrix whose columns are Ab1...Abp. That is, AB = A[b1...bp] =
[Ab1...Abp]
Each column of AB is a linear combination of the columns of A using weights from the corresponding
column of B.
3.3, 4.1-4.3, 4.5 QUESTIONS AND
CORRECT ANSWERS
2.1 Theorem 3 ✅✅ANSW-Let A and B denote matrices whose sizes are appropriate for the
following sums and products.
a. (A^T)^T = A
b. (A + B)^T = A^T + B^T
c. For any scalar r, (rA)^T = rA^T
d. (AB)^T = B^T * A^T
The transpose of a product of matrices equals the product of their transposes in reverse order.
2.2 Theorem 5 ✅✅ANSW-If A is an invertible n x n matrix then for each b in R^n, the equation Ax
= b has the unique solution x = A^-1 * b
2.2 Theorem 6 ✅✅ANSW-a. If A is an invertible matrix, then A^-1 is invertible and the inverse of
it is A.
b. If A and B are n x n matrices, then so is AB, and the inverse of AB is the product of the inverses of
A and B in the reverse order. That is:
(AB)^-1 = B^-1 * A^-1
c. If A is an invertible matrix, then so is A^T, and the inverse of A^T is the transpose of A^-1. That is:
(A^T)^-1 = (A^-1)^T
2.2 Theorem 7 ✅✅ANSW-An n x n matrix A is invertible if and 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
2.3 Theorem 8 ✅✅ANSW-Let A be a square n x n matrix. Then the following statements are
equivalent. That is, for a given A, the statements are either all true or all false.
, a. A is an invertible matrix.
b. A is row equivalent to the n x n identity matrix.
c. A has n pivot positions.
d. The equation Ax = 0 has only the trivial solution.
e. The columns of A for a linearly independent set.
f. The linear transformation x to Ax maps R^n onto R^n
g. The equation Ax = b has at least on solution for each b in R^n
h. The columns of A span R^n
i. The linear transformation x to Ax is one-to-one
j. There is an n x n matrix C such that CA = I
k. There is an n x n matrix D such that AD = I
A^T is an invertible matrix.
2.3 Theorem 9 ✅✅ANSW-Let T : R^n to R^n be a linear transformation and let A be the standard
matrix for T. Then T is invertible if and only if A is an invertible matrix. In that case, the linear
transformation S given by S(x) = A^-1 * x is the unique function satisfying equations (1) and (2),
2.1 Theorem 1 ✅✅ANSW-Let A, B, and C be matrices of the same size and let r and s be scalars.
a. A + B = B + A
b. (A + B) + C = A + (B + C)
c. A + 0 = A
d. (r(A + B) = rA + rB
e. (r + s)A = rA + sA
f. r(sA) = (rs)A
2.1 Definition ✅✅ANSW-If A is an m x n matrix, and if B is an n x p matrix with columns b1 ... bp,
then the product AB is the m x p matrix whose columns are Ab1...Abp. That is, AB = A[b1...bp] =
[Ab1...Abp]
Each column of AB is a linear combination of the columns of A using weights from the corresponding
column of B.
