Understanding Matrix Operations: Defined or Undefined?
This problem asks us to determine which statement regarding operations between two matrices, A and B, is incorrect. Let's first identify the dimensions of the given matrices.
Matrix A is given as:
$ A = \begin{bmatrix} 1 & 2 & -1 \\ 3 & 4 & 2 \\ 2 & 0 & 1 \end{bmatrix} $Matrix A has 3 rows and 3 columns, so its dimensions are $3 \times 3$.
Matrix B is given as:
$ B = \begin{bmatrix} 1 & 3 \\ -4 & 0 \\ 2 & 5 \end{bmatrix} $Matrix B has 3 rows and 2 columns, so its dimensions are $3 \times 2$.
Analyzing Matrix Compatibility for Operations
To determine if matrix operations are defined, we need to check the compatibility rules based on their dimensions.
Matrix Multiplication Rules:
- AB is defined if the number of columns in matrix A equals the number of rows in matrix B.
- BA is defined if the number of columns in matrix B equals the number of rows in matrix A.
Matrix Addition/Subtraction Rules:
- A + B or A - B are defined only if matrix A and matrix B have the exact same dimensions (same number of rows and same number of columns).
Evaluating the Given Options
Now, let's apply these rules to the matrices A ($3 \times 3$) and B ($3 \times 2$).
Option 1: AB is defined
Check compatibility for AB:
- Number of columns in A = 3
- Number of rows in B = 3
Since the number of columns in A (3) is equal to the number of rows in B (3), the matrix multiplication AB is defined. This statement is correct.
Option 2: BA is not defined
Check compatibility for BA:
- Number of columns in B = 2
- Number of rows in A = 3
Since the number of columns in B (2) is not equal to the number of rows in A (3), the matrix multiplication BA is not defined. This statement is correct.
Option 3: A + B is not defined
Check compatibility for A + B:
- Dimensions of A = $3 \times 3$
- Dimensions of B = $3 \times 2$
The dimensions of A and B are different. Therefore, the matrix addition A + B is not defined. This statement is correct.
Option 4: A - B is defined
Check compatibility for A - B:
- Dimensions of A = $3 \times 3$
- Dimensions of B = $3 \times 2$
The dimensions of A and B are different. Therefore, the matrix subtraction A - B is not defined. The statement claims that 'A - B is defined', which contradicts the rules of matrix operations. This statement is incorrect.
Conclusion
Based on the analysis, the statement "A - B is defined" is the one that is incorrect because matrices A and B do not have the same dimensions required for subtraction.
Important Questions from Miscellaneous
- Which of the following scheduler/schedulers is/are also called CPU scheduler ?
(A). Short Term Scheduler
(B). Long Term Scheduler
(C). Medium Term Scheduler
(D). Asymmetric Scheduler
Choose the correct answer from the options given below: - A situation where two or more processes are blocked, waiting for resources held by each other is called:
- External fragmentation occurs ________.
- Which disk scheduling algorithm looks for the track closest to the current head position?
- Which CPU scheduling algorithm prefers the process with the shortest burst time?
