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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- ‘Deep Water’ and ‘Indigo’ bring out the importance of overcoming fear. Discuss the two texts with reference to the above statement.
- CBSE CLASS XII - 2025
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- Identify A and B in the following reactions:
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- Alexia Limited invited applications for issuing 1,00,000 equity shares of ₹ 10 each at a premium of ₹ 10 per share. The amount was payable as follows :
On application ₹ 9 per share (including premium ₹ 6 per share)
On allotment ₹ 8 per share (including premium ₹ 4 per share)
On first and final call ₹ 3 per share.
Applications were received for 1,50,000 equity shares and allotment was made to the applicants as follows :
Category A : Applicants for 90,000 shares were allotted 70,000 shares.
Category B : Applicants for 60,000 shares were allotted 30,000 shares.
Excess money received on application was adjusted towards allotment and first and final call.
Shekhar, who had applied for 1,200 shares, failed to pay the first and final call. Shekhar belonged to Category B.
Pass necessary journal entries for the above transactions in the books of Alexia Limited. Open calls in arrears and calls in advance account wherever necessary.- CBSE CLASS XII - 2025
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- Premier Ltd. forfeited 600 shares of ₹ 10 each issued at a premium of ₹ 3 per share (payable with allotment) for non-payment of allotment money of ₹ 7 per share including premium. The first and final call of ₹ 3 per share was not yet made. The forfeited shares were reissued at ₹ 13 per share fully paid up.
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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