Register

GO Classes 2024 | Weekly Quiz 7 | Linear Algebra | Question: 19

• edited by
1,208 views
19 votes
19 votes

Suppose that we are solving $A x=\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right)$

In each of the option below, a complete solution $x$ is proposed.

Which of the following could possibly be the solution for above system of linear equations?

  1. $\vec{x}=\left(\begin{array}{l}
    1 \\
    2 \\
    3 \\
    4
    \end{array}\right)$
  2. $\vec{x}=\left(\begin{array}{l}1 \\ 2 \\ 3 \\ 4\end{array}\right)+\alpha_1\left(\begin{array}{c}1 \\ -1 \\ 5 \\ 17\end{array}\right)+\alpha_2\left(\begin{array}{l}1 \\ 0 \\ 0 \\ 1\end{array}\right)$ for all real numbers $\alpha_1, \alpha_2 \in \mathbb{R}$
  3. $\vec{x}=\left(\begin{array}{l}1 \\ 2\end{array}\right)+\alpha\left(\begin{array}{l}1 \\ 2\end{array}\right)$ for all real numbers $\alpha \in \mathbb{R}$
  4. $\vec{x}=\left(\begin{array}{l}1 \\ 2\end{array}\right)+\alpha\left(\begin{array}{c}1 \\ -1\end{array}\right)$ for all real numbers $\alpha \in \mathbb{R}$
• edited by
User Avatar

1 Answer

15 votes
15 votes
B, D.

Option A)

The dimension of x and b gives the dimension of matrix A.

For this option, the Dimension of matrix A must be 3 * 4.

Since the number of rows is less than the number of columns, the matrix may have zero solutions OR infinite solutions.

Option A is just a single solution, Therefore It is false.

Option B)
From the similar reasoning above, option B is a possible solution since it is infinite solution for Ax = $\begin{bmatrix} 1\\ 2\\ 3 \end{bmatrix}$.

Option C)

For this option, the dimension of the matrix must be 3 * 2.

In the complete solution, it is given that $\begin{bmatrix} 1\\ 2\end{bmatrix}$ solves Ax = $\begin{bmatrix} 1\\ 2\\ 3 \end{bmatrix}$.

It is also given that  $\begin{bmatrix} 1\\ 2\end{bmatrix}$  solves Ax = 0. It is a contradiction and therefore, option C is false.

Option D)

From the similar reasoning of option C, it is possible for option D to be true.
• edited by
User Avatar
Voting & Accepting an answer helps improve the community
Answer:
← PreviousNext →
← Previous in categoryNext in category →

Related questions

1.2k
views
4 answers
10 votes
GO Classes asked Apr 5, 2023
1,156 views
GO Classes 2024 | Weekly Quiz 7 | Linear Algebra | Question: 6
If $\text{A}$ is a $4 \times 3$ matrix and $\text{A} x=b$ is not solvable for some $b$ and the solutions are not unique when they exist, possible values for the rank of $\text{A}$ are ________ (list all possibilities).$0$1$2$3$
User Avatar
736
views
1 answers
15 votes
GO Classes asked Apr 5, 2023
736 views
GO Classes 2024 | Weekly Quiz 7 | Linear Algebra | Question: 7
$A x=b$ has solutions $x_1=\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right)$ and $x_2=\left(\begin{array}{l}4 \\ 5 \\ 6\end{array}\right)$, and possibly ... \\3\end{array}\right)$\left(\begin{array}{l}4 \\4 \\4\\\end{array} \right)$
User Avatar
928
views
2 answers
11 votes
GO Classes asked Apr 5, 2023
928 views
GO Classes 2024 | Weekly Quiz 7 | Linear Algebra | Question: 5
Suppose that the characteristic polynomial of $\text{A}$ is$p(\lambda)=\lambda(\lambda-2)(\lambda-3)^2.$Which of the following can you determine from this ... is symmetric.Whether $\text{A}$ is diagonalizable.The eigenvalues of $\text{A}$.
User Avatar
1.0k
views
2 answers
14 votes
GO Classes asked Apr 5, 2023
1,043 views
GO Classes 2024 | Weekly Quiz 7 | Linear Algebra | Question: 8
Suppose that $A$ is a $3 \times 3$ real-symmetric matrix with eigenvalues $\lambda_1=1$, $\lambda_2=-1, \lambda_3=-2$, and corresponding eigenvectors $x_1, x_2, x_3.$You ... }\right)$x_3 =\left(\begin{array}{c}0 \\0 \\0\end{array}\right)$
User Avatar
Link Copied!