If A'= \(\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 &1 \end{bmatrix}\)\(\begin{bmatrix} -1 & 2 & 1 \\ 1 &2 & 3\end{bmatrix}\) , then verify that
(i) \((A+B)'=A'+B' \)
(ii) \((A-B)'=A'-B'\)
(i) \((A+B)'=A'+B' \)
(ii) \((A-B)'=A'-B'\)
Solution and Explanation
(i) It is known that A=(A')'
Therefore, we have:
A= \(\begin{bmatrix} 3 & -1 & 0 \\ 4 & 2 & 1 \end{bmatrix}\)
B'= \(\begin{bmatrix} -1 & 1 \\ 2 & 2 \\ 1 &3 \end{bmatrix}\)
\(A+B\) = \(\begin{bmatrix} 3 & -1 & 0 \\ 4 & 2 & 1 \end{bmatrix}\) + \(\begin{bmatrix} -1 & 2 & 1 \\ 1 &2 & 3\end{bmatrix}\)= \(\begin{bmatrix} 2 & 1 & 1 \\ 5 & 4 & 4 \end{bmatrix}\)
\(\therefore (A+B)'=\) \(\begin{bmatrix} 2 & 5 \\ 1 & 4 \\ 1 &4 \end{bmatrix}\)
\(A'+B'=\) \(\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 &1 \end{bmatrix}\)+ \(\begin{bmatrix} -1 & 1 \\ 2 & 2 \\ 1 &3 \end{bmatrix}\)= \(\begin{bmatrix} 2 & 5 \\ 1 & 4 \\ 1 &4 \end{bmatrix}\)
Thus, we verified that:(A+B)'=A'+B'
(ii) \(A-B\)= \(\begin{bmatrix} 3 & -1 & 0 \\ 4 & 2 & 1 \end{bmatrix}\)- \(\begin{bmatrix} -1 & 2 & 1 \\ 1 &2 & 3\end{bmatrix}\) = \(\begin{bmatrix} 4 & -3 & -1 \\ 3 &0 & -2\end{bmatrix}\)
so\( (A-B)'\) = \(\begin{bmatrix} -4 & 3 \\ -3 & 0 \\ -1 &-2 \end{bmatrix}\)
A'-B'= \(\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 &1 \end{bmatrix}\)- \(\begin{bmatrix} -1 & 1 \\ 2 & 2 \\ 1 &3 \end{bmatrix}\)= \(\begin{bmatrix} -4 & 3 \\ -3 & 0 \\ -1 &-2 \end{bmatrix}\)
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