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.\(\begin{bmatrix}-1&5&6\\\sqrt3&5&6\\2&3&-1\end{bmatrix}\)
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.\(\begin{bmatrix}-1&5&6\\\sqrt3&5&6\\2&3&-1\end{bmatrix}\)
Solution and Explanation
(i) Let A=\(\begin{bmatrix}5\\\frac{1}{2}\\-1\end {bmatrix}\)
then A-T= \(\begin{bmatrix}5&\frac{1}{2}&-1\end{bmatrix}\)
(ii)Let A= \(\begin{bmatrix}1&-1\\2&3\end{bmatrix}\)
then A-T= \(\begin{bmatrix}1&2\\-1&3\end{bmatrix}\)
(iii)Let A= \(\begin{bmatrix}-1&5&6\\\sqrt3&5&6\\2&3&-1\end{bmatrix}\)
then A-T = \(\begin{bmatrix}-1&\sqrt3&2\\5&5&3\\6&6&-1\end{bmatrix}\)
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Concepts Used:
Transpose of a Matrix
The matrix acquired by interchanging the rows and columns of the parent matrix is called the Transpose matrix. The transpose matrix is also defined as - “A Matrix which is formed by transposing all the rows of a given matrix into columns and vice-versa.”
The transpose matrix of A is represented by A’. It can be better understood by the given example:
Now, in Matrix A, the number of rows was 4 and the number of columns was 3 but, on taking the transpose of A we acquired A’ having 3 rows and 4 columns. Consequently, the vertical Matrix gets converted into Horizontal Matrix.
Hence, we can say if the matrix before transposing was a vertical matrix, it will be transposed to a horizontal matrix and vice-versa.
Read More: Transpose of a Matrix