If A is a 5×6 matrix and Z is a 6 X5 zero matrix then A + Zt = A Select one:
A True
B False
The Correct Answer and Explanation is :
The correct answer is A: True.
Explanation:
Let’s break down the problem step by step.
- Matrix A is a (5 \times 6) matrix, which means it has 5 rows and 6 columns.
- Matrix Z is a (6 \times 5) zero matrix, which means it has 6 rows and 5 columns and all its entries are zero.
The operation (Z^T) (transpose of Z):
- The transpose of a matrix involves switching its rows and columns. So, the transpose of (Z), denoted (Z^T), will be a (5 \times 6) matrix.
- Since (Z) is a zero matrix, all entries in (Z) are zero. Therefore, (Z^T) will also be a zero matrix with the same dimensions, i.e., a (5 \times 6) zero matrix.
Matrix addition (A + Z^T):
- Matrix (A) is a (5 \times 6) matrix, and (Z^T) is also a (5 \times 6) matrix (as we saw above). Since both matrices have the same dimensions, we can add them element-wise.
- Since (Z^T) is a zero matrix, adding it to (A) will not change the values of the entries in (A). The addition would look like:
[
A + Z^T = A + \text{(a matrix of zeros)} = A
]
Therefore, the expression (A + Z^T) equals matrix (A), which makes the statement True.
Conclusion:
Since adding the zero matrix (which is the transpose of (Z)) to (A) doesn’t change (A), the correct answer is A: True.