If A=\(\begin{bmatrix}-1&2&3\\5&7&9\\-2&1&1\end{bmatrix}\)and 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=\(\begin{bmatrix}-1&2&3\\5&7&9\\-2&1&1\end{bmatrix}\)and 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'
Solution and Explanation
We have: A'=\(\begin{bmatrix}-1&5&-2\\2&7&1\\3&9&1\end{bmatrix}\),B'=\(\begin{bmatrix}-4&1&1\\1&2&3\\-5&0&1\end{bmatrix}\)
(i)A+B= \(\begin{bmatrix}-1&2&3\\5&7&9\\-2&1&1\end{bmatrix}\)+\(\begin{bmatrix}-4&1&-5\\1&2&0\\1&3&1\end{bmatrix}\)
=\(\begin{bmatrix}-5&3&-2\\6&9&9\\1&4&2\end{bmatrix}\)
therefore (A+B)'=\(\begin{bmatrix}-5&6&-1\\3&9&4\\-2&9&2\end{bmatrix}\)
A'+B'=\(\begin{bmatrix}-1&5&-2\\2&7&1\\3&9&1\end{bmatrix}\)+\(\begin{bmatrix}-4&1&1\\1&2&3\\-5&0&1\end{bmatrix}\)
Hence we verified that (A+B)'=A'+B'
(ii)A-B=\(\begin{bmatrix}-1&2&3\\5&7&9\\-2&1&1\end{bmatrix}\)-\(\begin{bmatrix}-4&1&-5\\1&2&0\\1&3&1\end{bmatrix}\)
=\(\begin{bmatrix}3&1&8\\4&5&9\\-3&-2&0\end{bmatrix}\)
therefore (A-B)'=\(\begin{bmatrix}-3&4&-3\\1&5&-2\\8&9&0\end{bmatrix}\)
A'-B'=\(\begin{bmatrix}-1&5&-2\\2&7&1\\3&9&1\end{bmatrix}\)-\(\begin{bmatrix}-4&1&1\\1&2&3\\-5&0&1\end{bmatrix}\)
=\(\begin{bmatrix}-3&4&-3\\1&5&-2\\8&9&0\end{bmatrix}\)
Hence we verified that (A-B)'=A'-B'
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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