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 relationships is/are not true?
(A). Most probable velocity = $\sqrt{\frac{2RT}{M}}$
(B). PV = $\frac{3}{2}kT$
(C). Compressibility factor Z = $\frac{pV}{nRT}$
(D). Average kinetic energy of gas = $\frac{1}{2}kT$
Choose the correct answer from the options given below - Match List-I with List-II
List-I List-II Electronic Configuration First Ionisation energy (kJ mol$^{-1}$) (A). ns$^2$ (I). 2100 (B). ns$^2$np$^1$ (II). 1400 (C). ns$^2$np$^3$ (III). 800 (D). ns$^2$np$^6$ (IV). 900
Choose the correct answer from the options given below: - The shielding constant of a 2p electron (calculated using Slater's rules) is
- Match List-I with List-II
List-I List-II Spectroscopy Property (A). Raman (I). Polarizability (B). FTIR (II). Dipole Moment (C). UV-Visible (III). Absorbance (D). NMR (IV). Spin
Choose the correct answer from the options given below: - The structure of protein comprises of:
(A). Primary structure of protein is associated with amino acids
(B). Secondary structure of protein is associated to peptides
(C). Tertiary structure of protein is associated with polypeptide chains
(D). Quaternary structure of protein is associated with polypeptide chains
Choose the correct answer from the options given below:
