Chapter 3, Problem 1SE

Mark each statement True or False. Justify each answer.

Assume that all matrices here are square.

1. a. If A is a 2 × 2 matrix with a zero determinant, then one column of A is a multiple of the other.
2. b.If two rows of a 3 × 3 matrix A are the same, then det A = 0.
3. c. If A is a 3 × 3 matrix, then det 5A = 5det A.
4. d.If A and B are n × n matrices, with det A = 2 and det B = 3, then det(A + B) = 5.
5. e. If A is n × n and det A = 2, then det A³ = 6.
6. f. If B is produced by interchanging two rows of A, then det B = det A.
7. g.If B is produced by multiplying row 3 of A by 5, then det B = 5·det A.
8. h.If B is formed by adding to one row of A a linear combination of the other rows, then det B = det A.
9. i. det A^T = − det A.
10. j. det(−A)= − det A.
11. k.det A^TA ≥ 0.
12. l. Any system of n linear equations in n variables can be solved by Cramer’s rule.
13. 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.
14. n. If A³ = 0. then det A = 0.
15. ○.  If A is invertible, then det A⁻¹ = det A.
16. p. If A is invertible, then (det A)(det A⁻¹) = 1.

To determine

To check: Whether the statement “If A is a $2\times 2$ matrix with a zero determinant, then one column of A is a multiple of the other” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Consider A be a $2\times 2$ matrix.

The $\det A=0$.

Then, the matrix A have linearly dependent columns.

Thus, the statement is true.

To determine

To check: Whether the statement “If two rows of a $3\times 3$ matrix A are the same, then $\det A=0$” is true or false.

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 $3\times 3$ matrix are same.

Then, the $\det A=0$, so the two rows of the matrix A are equal.

Therefore, the statement is true.

To determine

To check: Whether the statement “If A is a $3\times 3$ matrix, then $\det 5A=5\det A$” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Consider A be a $3\times 3$ matrix.

The properties of the determinant as follows.

$\det dA=d^n\left(\det A\right)$ (1)

Substitute 3 for n and 5 for d in equation (1),

$\det 5A=5^3\left(\det A\right)$

Thus, the statement is false.

To determine

To check: Whether the statement “If A and B are $n\times n$ matrices, with $\det A=2$ and $\det B=3$, then $\det \left(A+B\right)=5$” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Consider the value of matrix A and B as follows.

$\begin{array}{l}A=\left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right]\\ B=\left[\begin{array}{cc}1& 0\\ 0& 3\end{array}\right]\end{array}$

Calculate the value of $\det A$.

$\begin{array}{c}\det A=2\times 1-0\times 0\\ =2\end{array}$

Calculate the value of $\det B$.

$\begin{array}{c}\det A=3\times 1-0\times 0\\ =3\end{array}$

Calculate $\left(\det A+\det B\right)$ as follows.

$\begin{array}{c}\left(\det A+\det B\right)=2+3\\ =5\end{array}$

Calculate the value of $\left(A+B\right)$ as follows.

$\begin{array}{c}A+B=\left[\begin{array}{cc}2& 0\\ 0& 1\end{array}\right]+\left[\begin{array}{cc}1& 0\\ 0& 3\end{array}\right]\\ =\left[\begin{array}{cc}3& 0\\ 0& 4\end{array}\right]\end{array}$

Calculate the value of $\det \left(A+B\right)$ as follows.

$\begin{array}{c}\det \left(A+B\right)=3\times 4-0\times 0\\ =12\end{array}$

Compare $\det \left(A+B\right)$ and $\left(\det A+\det B\right)$ as follows.

$\begin{array}{c}\det \left(A+B\right)\ne \left(\det A+\det B\right)\\ 12\ne 5\end{array}$

Thus, the statement is false.

To determine

To check: Whether the statement “If A is $n\times n$ and $\det A=2$, then $\det A^3=6$” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Theorem used:

“If A and B are $n\times n$ matrices, then $\det AB=\left(\det A\right)\left(\det B\right)$”.

Calculation:

The value of $\det A=2$.

Calculate value of $\det A^3$ by using the above theorem.

$\begin{array}{c}\det A^3=\det \left(A\times A\times A\right)\\ =\left(\det A\right)\times \left(\det A\right)\times \left(\det A\right)\\ =2\times 2\times 2\\ =8\ne 6\end{array}$

Thus, the statement is false.

To determine

To check: Whether the statement “If B is produced by interchanging two rows of A, then $\det B=\det A$” is true or false.

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 $\det B=\det A$.

b. If two rows of matrix A are interchanged to produce matrix B, then, $\det A=-\det B$.

c. If one row of A is multiplied by k to produce B, then $\det B=k\cdot \det A$”.

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 $\det A=-\det B$.

Hence, the statement is false.

To determine

To check: Whether the statement “If B is produced by multiplying row 3 of A by 5, then

$\det B=5\det A$” is true or false.

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 $\det B=5\det A$.

Thus, the given statement is true.

To determine

To check: Whether the statement “If B is formed by adding to one row of A a linear combination of the other rows, then $\det B=\det A$” is true or false.

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 $\det B=\det A$.

Thus, the given statement is true.

To determine

To check: Whether the statement “$\det A^T=-\det A$” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Theorem used:

“Consider A is a $n\times n$ matrix, then $\det A^T=\det A$”.

From above theorem, $\det A^T\ne -\det A$.

Thus, the given statement is false.

To determine

To check: Whether the statement “$\det \left(-A\right)=-\det A$” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

From part (c) of above theorem,  the statement $\det \left(-A\right)=-\det A$ is false.

Thus, the given statement is false.

To determine

To check: Whether the statement “$\det A^{T}A\ge 0$” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

From the above theorem, $\det A=\det A^T$

Calculate the value of $\left(\det A^{T}A\right)$ as follows.

$\left(\det A^{T}A\right)=\left(\det A^T\right)\left(\det A\right)$

Substitute A for $A^T$,

$\begin{array}{c}\left(\det A{A}^T\right)=\left(\det A\right)\left(\det A\right)\\ ={\left(\det A\right)}^2\end{array}$

Thus, $\left(\det A^{T}A\right)\ge 0$.

Thus, the given statement is true.

To determine

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 $n\times n$ matrix. For any b in ${ℝ}^n$, the unique solution x of $A\text{x}=\text{b}$ has entries given by,

$x_i=\frac{\det A_i\left(\text{b}\right)}{\det A}$”.

Here, the value of i ranges from $i=1,2,3,\mathrm{...},n$.

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.

To determine

To check: Whether the statement “If u and v are in ${ℝ}^n$ and , then the area of the triangle in the plane with vertices at 0, u, and v is 10” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Theorem used:

“If A is a $2\times 2$ matrix, the area of the parallelogram determined by the columns of A is $\left|\det A\right|$. If A is a $3\times 3$ matrix, the volume of the parallelepiped determined by the columns of A is $\det A$.”

Consider u and v are in ${ℝ}^2$.

Show the value of $\det \left[\begin{array}{cc}\text{u}& \text{v}\end{array}\right]$ as follows.

$\det \left[\begin{array}{cc}\text{u}& \text{v}\end{array}\right]=10$

Calculate area of the triangle by using the above theorem.

$\begin{array}{c}\text{Area}\text{of}\text{triangle}=\frac{1}{2}\left(\text{Area}\text{of}\text{parallel}\text{ogram}\right)\\ =\frac{1}{2}\left|\det A\right|\end{array}$

Substitute 10 for $\det A$,

$\begin{array}{c}\text{Area}\text{of}\text{triangle}=\frac{1}{2}\left|10\right|\\ =5\end{array}$

Thus, the given statement is false.

To determine

To check: Whether the statement “If $A^3=0$, then $\det A=0$” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Calculate the value of $\det A^3$ as follows by using the above theorem.

$\begin{array}{c}\det A^3=\det \left(A\times A\times A\right)\\ =\left(\det A\right)\left(\det A\right)\left(\det A\right)\\ ={\left(\det A\right)}^3\end{array}$

Thus, the given statement is true.

To determine

To check: Whether the statement “If A is invertible, then $\det A^{-1}=\det A$” is true or false.

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,

$\det A^{-1}=\frac{1}{\det A}$.

Thus, the given statement is false.

To determine

To check: Whether the statement “If A is invertible, then $\left(\det A\right)\left(\det A^{-1}\right)=1$” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Consider A as invertible matrix.

Calculate the value of $\left(\det A\right)\left(\det A^{-1}\right)$ by using the above theorem.

$\begin{array}{c}\left(\det A\right)\left(\det A^{-1}\right)=\det \left(A\times A^{-1}\right)\\ =\det \left(I\right)\\ =I\end{array}$