
Mark each statement True or False. Justify each answer.
Assume that all matrices here are square.
- a. If A is a 2 × 2 matrix with a zero determinant, then one column of A is a multiple of the other.
- b.If two rows of a 3 × 3 matrix A are the same, then det A = 0.
- c. If A is a 3 × 3 matrix, then det 5A = 5det A.
- d.If A and B are n × n matrices, with det A = 2 and det B = 3, then det(A + B) = 5.
- e. If A is n × n and det A = 2, then det A3 = 6.
- f. If B is produced by interchanging two rows of A, then det B = det A.
- g.If B is produced by multiplying row 3 of A by 5, then det B = 5·det A.
- h.If B is formed by adding to one row of A a linear combination of the other rows, then det B = det A.
- i. det AT = − det A.
- j. det(−A)= − det A.
- k.det ATA ≥ 0.
- l. Any system of n linear equations in n variables can be solved by Cramer’s rule.
- m. If u and v are in ℝ2 and det [u v] = 10, then the area of the
triangle in the plane with vertices at 0, u, and v is 10. - n. If A3 = 0. then det A = 0.
- ○. If A is invertible, then det A−1 = det A.
- p. If A is invertible, then (det A)(det A−1) = 1.
a.

To check: Whether the statement “If A is a
Answer to Problem 1SE
The given statement is true.
Explanation of Solution
Consider A be a
The
Then, the matrix A have linearly dependent columns.
Thus, the statement is true.
b.

To check: Whether the statement “If two rows of a
Answer to Problem 1SE
The given statement is true.
Explanation of Solution
From problem 30 in section 3.2, it is clear that the two rows of
Then, the
Therefore, the statement is true.
c.

To check: Whether the statement “If A is a
Answer to Problem 1SE
The given statement is false.
Explanation of Solution
Consider A be a
The properties of the determinant as follows.
Substitute 3 for n and 5 for d in equation (1),
Thus, the statement is false.
d.

To check: Whether the statement “If A and B are
Answer to Problem 1SE
The given statement is false.
Explanation of Solution
Consider the value of matrix A and B as follows.
Calculate the value of
Calculate the value of
Calculate
Calculate the value of
Calculate the value of
Compare
Thus, the statement is false.
e.

To check: Whether the statement “If A is
Answer to Problem 1SE
The given statement is false.
Explanation of Solution
Theorem used:
“If A and B are
Calculation:
The value of
Calculate value of
Thus, the statement is false.
f.

To check: Whether the statement “If B is produced by interchanging two rows of A, then
Answer to Problem 1SE
The given statement is false.
Explanation of Solution
Theorem used:
“Let A be a square matrix.
a. If a multiple of one row of A is added to another row to produce a matrix B, then
b. If two rows of matrix A are interchanged to produce matrix B, then,
c. If one row of A is multiplied by k to produce B, then
From the above theorem part (b), it is clear that the 2 rows of the matrix A are interchanged to get matrix B.
Then, there occurs the determinant value as
Hence, the statement is false.
g.

To check: Whether the statement “If B is produced by multiplying row 3 of A by 5, then
Answer to Problem 1SE
The given statement is true.
Explanation of Solution
From the above theorem in part (c), “If B is produced by multiplying row 3 of A by 5”, then
Thus, the given statement is true.
h.

To check: Whether the statement “If B is formed by adding to one row of A a linear combination of the other rows, then
Answer to Problem 1SE
The given statement is true.
Explanation of Solution
From part (a) of above theorem, “If B is formed by adding to one row of A a linear combination of the other rows”, then
Thus, the given statement is true.
i.

To check: Whether the statement “
Answer to Problem 1SE
The given statement is false.
Explanation of Solution
Theorem used:
“Consider A is a
From above theorem,
Thus, the given statement is false.
j.

To check: Whether the statement “
Answer to Problem 1SE
The given statement is false.
Explanation of Solution
From part (c) of above theorem, the statement
Thus, the given statement is false.
k.

To check: Whether the statement “
Answer to Problem 1SE
The given statement is true.
Explanation of Solution
From the above theorem,
Calculate the value of
Substitute A for
Thus,
Thus, the given statement is true.
l.

To check: Whether the statement “Any system of n linear equations in n variables can be solved by Cramer’s rule” is true or false.
Answer to Problem 1SE
The given statement is false.
Explanation of Solution
Theorem 7: “Let A be an invertible
Here, the value of i ranges from
From above Theorem 7, the statement “Any system of n linear equations in n variables can be solved by Cramer’s rule’’ is false as the coefficient matrix has to be invertible.
Thus, the given statement is false.
m.

To check: Whether the statement “If u and v are in
Answer to Problem 1SE
The given statement is false.
Explanation of Solution
Theorem used:
“If A is a
Consider u and v are in
Show the value of
Calculate area of the triangle by using the above theorem.
Substitute 10 for
Thus, the given statement is false.
n.

To check: Whether the statement “If
Answer to Problem 1SE
The given statement is true.
Explanation of Solution
Calculate the value of
Thus, the given statement is true.
o.

To check: Whether the statement “If A is invertible, then
Answer to Problem 1SE
The given statement is false.
Explanation of Solution
Refer Exercise 31 of Section 3.2.
Consider A as invertible matrix. Then,
Thus, the given statement is false.
p.

To check: Whether the statement “If A is invertible, then
Answer to Problem 1SE
The given statement is true.
Explanation of Solution
Consider A as invertible matrix.
Calculate the value of
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