Question: Write down \(\mathrm{T}\) if the following statement is true and \(\mathrm{F}\) if the statement is false. ( 2 points each)

1. The rank of a \(5 \times 4\) matrix could be odd.
2. If \(\mathrm{A}\) is a \(3 \times 3\) matrix with rank 3, then the equation \(A \vec{x}=\vec{b}\) always has a unique solution.
3. Let \(A\) be an \(n \times p\) matrix and \(B\) be a \(p \times m\) matrix. Then the kernel of $A B$ is always contained in the kernel of \(A\).
4. The set of vectors of the form \(\left[\begin{array}{l}a \\ 0 \\ b \\ b\end{array}\right]\) is a linear subspace in \(\mathbb{R}^{4}\).
5. The transformation \(T(A)=2 A\) from \(\mathbb{R}^{5 \times 5}\) to \(\mathbb{R}^{5 \times 5}\) is an isomorphism.
6. If \(A\) is an orthogonal matrix, then \(A^{-1}\) must also be orthogonal.
7. If \(A\) is not the zero matrix, then at least one of \(A\) and \(-A\) must have a negative singular value.
8. If \(A\) is a symmetric \(7 \times 7\) matrix \(\left(A^{T}=A\right)\), then the determinant of \(A\) is positive.
9. If every singular value of \(A\) is 1, then \(A\) is orthogonal.
10. If \(A\) is symmetric and \(S\) is orthogonal, then \(S A S^{-1}\) is symmetric as well.
11. If \(A\) is a skew-symmetric matrix \(\left(A^{T}=-A\right)\), then the determinant of \(A\) must be negative.
12. The following matrix is positive definite:
    \[\left[\begin{array}{lll} 2 & 1 & 0 \\ 1 & 3 & 1 \\ 0 & 1 & 2 \end{array}\right]\]
13. If \(\lambda\) is an eigenvalue of \(A\), then \(\lambda\) is also a singular value of \(A\).
14. Every \(3 \times 3\) matrix has at least one real eigenvalue.
15. If \(u\) and \(v\) are linearly independent eigenvectors, then they correspond to different eigenvalues.

Price: $2.99

Solution: The downloadable solution consists of 2 pages
Deliverable: Word Document