Find the transpose of each of the following matrices:
(i) \(\begin{bmatrix} 5 \\ \frac{1}{2} \\ -1 \end{bmatrix}\)
(ii) \(\begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix}\)
(iii) \(\begin{bmatrix} -1 & 5 & 6 \\ \sqrt{3} & 5 & 6 \\ 2 & 3 & -1 \end{bmatrix}\)
(i) \(\begin{bmatrix} 5 & \frac{1}{2} & -1 \end{bmatrix}\)
(ii) \(\begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix}\)
(iii) \(\begin{bmatrix} -1 & \sqrt{3} & 2 \\ 5 & 5 & 3 \\ 6 & 6 & -1 \end{bmatrix}\)
If
\[ A = \begin{bmatrix} -1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} -4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1 \end{bmatrix} \]
then verify that
(i) \((A + B)' = A' + B'\)
(ii) \((A - B)' = A' - B'\)
If
\[ A' = \begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix} \]
then verify that
(i) \((A + B)' = A' + B'\)
(ii) \((A - B)' = A' - B'\)
If
\[ A' = \begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix} \]
then find \((A + 2B)'\).
\((A + 2B)' = \begin{bmatrix} -4 & 5 \\ 1 & 6 \end{bmatrix}\)
For the matrices \(A\) and \(B\), verify that \((AB)' = B'A'\), where
(i) \(A = \begin{bmatrix} 1 \\ -4 \\ 3 \end{bmatrix},\; B = \begin{bmatrix} -1 & 2 & 1 \end{bmatrix}\)
(ii) \(A = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix},\; B = \begin{bmatrix} 1 & 5 & 7 \end{bmatrix}\)
If
(i) \(A = \begin{bmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{bmatrix}\), then verify that \(A'A = I\).
(ii) If \(A = \begin{bmatrix} \sin\alpha & \cos\alpha \\ -\cos\alpha & \sin\alpha \end{bmatrix}\), then verify that \(A'A = I\).
(i) Show that the matrix
\[ A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix} \]
is a symmetric matrix.
(ii) Show that the matrix
\[ A = \begin{bmatrix} 0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0 \end{bmatrix} \]
is a skew symmetric matrix.
For the matrix \(A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}\), verify that
(i) \((A + A')\) is a symmetric matrix
(ii) \((A - A')\) is a skew symmetric matrix
Find \(\frac{1}{2}(A + A')\) and \(\frac{1}{2}(A - A')\), when
\[ A = \begin{bmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{bmatrix} \]
\(\frac{1}{2}(A + A') = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\)
\(\frac{1}{2}(A - A') = \begin{bmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{bmatrix}\)
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
(i) \(\begin{bmatrix} 3 & 5 \\ 1 & -1 \end{bmatrix}\)
(ii) \(\begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix}\)
(iii) \(\begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix}\)
(iv) \(\begin{bmatrix} 1 & 5 \\ -1 & 2 \end{bmatrix}\)
(i) \(A = \begin{bmatrix} 3 & 3 \\ 3 & -1 \end{bmatrix} + \begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}\)
(ii) \(A = \begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix} + \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\)
(iii) \(A = \begin{bmatrix} 3 & \frac{1}{2} & -\frac{5}{2} \\ \frac{1}{2} & -2 & -2 \\ -\frac{5}{2} & -2 & 2 \end{bmatrix} + \begin{bmatrix} 0 & \frac{5}{2} & \frac{3}{2} \\ -\frac{5}{2} & 0 & 3 \\ -\frac{3}{2} & -3 & 0 \end{bmatrix}\)
(iv) \(A = \begin{bmatrix} 1 & 2 \\ 2 & 2 \end{bmatrix} + \begin{bmatrix} 0 & 3 \\ -3 & 0 \end{bmatrix}\)
Choose the correct answer.
If \(A, B\) are symmetric matrices of same order, then \(AB - BA\) is a
Skew symmetric matrix
Symmetric matrix
Zero matrix
Identity matrix
Choose the correct answer.
If \(A = \begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix}\), and \(A + A' = I\), then the value of \(\alpha\) is
\(\frac{\pi}{6}\)
\(\frac{\pi}{3}\)
\(\pi\)
\(\frac{3\pi}{2}\)