Math 240 Sections 2.1-2.3, 2.6, 3.1-3.3,
4.1-4.3, 4.5
2.1 Theorem 1 - ANS-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 - ANS-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.
2.1 Theorem 2 - ANS-Let A be an m x n matrix and let B and C have sizes for which the
indicated sums and products are defined.
a. A(BC) = (AB)C
b. A(B+C) = AB + AC
c. (B + C)A = BA + CA
d. r(AB) = (rA)B = (rB)A for any scalar r
e. Im * A = A = A * In
2.1 Theorem 3 - ANS-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 - ANS-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 - ANS-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 - ANS-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 - ANS-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 - ANS-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),
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