Description
1-1: Find the solution set of the linear system
x + 2y + 3z = 1
2x + 4y + 7z = 2
3x + 7y + 11z = 8
1-2: (1.1 no. 31) Find the polynomial f(t) = a + bt + ct2 of degree 2 whose graph passes through the
points (1, −1), (2, 3), and (3, 13).
1-3: Consider the linear system (1.1 no. 19)
x + y − z = −2
3x − 5y + 13z = 18
x − 2y + 5z = k
where k is an arbitrary number.
(a) For which value(s) of k does this system have solution(s)?
(b) Find all solutions for each value of k found in part (a).
1-4: A three-digit number has two properties. The tens-digit and the ones-digit add up to 5. If the number
is written with the digits in the reverse order, and then subtracted from the original number, the
result is 792. Use a system of equations to find all of the three-digit numbers with these properties.
1-5: Let J be an m × n matrix with all entries equal to 1, i.e., (J)ij = 1 for 1 ≤ i ≤ m and 1 ≤ j ≤ n.
Let B be an m × n matrix with
B =
1 2 3 · · · n − 1 n
1 2 3 · · · n − 1 n
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1 2 3 · · · n − 1 n
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(a) Find J
T B.
(b) Find BJT
.
1-6: Let A be an m × n matrix. Show that AAT
is a symmetric matrix.
1-7: Let A be an n × n matrix. Prove that there exists a symmetric matrix X and a skew-symmetric
matrix Y such that A = X + Y .
