Let A and B be non-singular matrices of the same order such that AB = A and BA = B. Which of the following statements is/are correct ? 1. A 2 = A 2. AB 2 = A 2B Select the correct answer using the code given below:
Both 1 and 2
Understanding Matrix Properties from Given Conditions
We are given two non-singular matrices, A and B, of the same order. Non-singular means that the inverse of each matrix exists, denoted as \( A^{-1} \) and \( B^{-1} \). We are also given two important conditions:
- Condition 1: \( AB = A \)
- Condition 2: \( BA = B \)
We need to determine which of the following statements are correct based on these conditions:
- \( A^2 = A \)
- \( AB^2 = A^2B \)
Analyzing the Given Conditions
Let's use the fact that A and B are non-singular. From Condition 1, \( AB = A \). Since A is non-singular, we can multiply by \( A^{-1} \) on the left side of the equation:
\( A^{-1}(AB) = A^{-1}A \)
\( (A^{-1}A)B = I \)
\( IB = I \)
\( B = I \)
Here, \( I \) represents the identity matrix of the same order as A and B.
Now let's use Condition 2, \( BA = B \). Since B is non-singular, we can multiply by \( B^{-1} \) on the right side of the equation:
\( (BA)B^{-1} = BB^{-1} \)
\( B(AB^{-1}) = I \)
Alternatively, and perhaps more directly, substitute the result \( B = I \) into Condition 2:
\( IA = I \)
\( A = I \)
So, from the given conditions and the fact that A and B are non-singular, we conclude that both matrices A and B must be the identity matrix \( I \).
Evaluating Statement 1: \( A^2 = A \)
We found that \( A = I \). Let's substitute this into the statement:
\( A^2 = I^2 \)
\( I^2 = I \)
So, \( A^2 = I \). Since \( A = I \), the statement \( A^2 = A \) becomes \( I = I \), which is true.
Therefore, Statement 1 is correct.
Evaluating Statement 2: \( AB^2 = A^2B \)
We found that \( A = I \) and \( B = I \). Let's substitute these into the statement:
Left Hand Side (LHS): \( AB^2 = I \cdot I^2 = I \cdot I = I \)
Right Hand Side (RHS): \( A^2B = I^2 \cdot I = I \cdot I = I \)
Since LHS = RHS (\( I = I \)), the statement \( AB^2 = A^2B \) is true when \( A = I \) and \( B = I \).
Therefore, Statement 2 is correct.
Conclusion
Based on the given conditions that A and B are non-singular matrices satisfying \( AB = A \) and \( BA = B \), we deduced that both matrices must be the identity matrix \( I \). Evaluating the two statements using \( A=I \) and \( B=I \) showed that both \( A^2=A \) and \( AB^2=A^2B \) are correct.
Thus, both Statement 1 and Statement 2 are correct.
Important Questions from Operations on Matrices
If \(A=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]\), then what is the value of det(I + AA'), where I is the 3 × 3 identity matrix?
If \(A=\left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{array}\right]\), then which of the following statements are correct?
1. An will always be singular for any positive integer n.
2. An will always be a diagonal matrix for any positive integer n.
3. An will always be a symmetric matrix for any positive integer n.
Select the correct answer using the code given below:
If \(A=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\), then what is 23A3 - 19A2 - 4A equal to ?
If A is an orthogonal matrix of order 3 and \({\rm{B}} = \left[ {\begin{array}{*{20}{c}} 1&2&3\\ { - 3}&0&2\\ 2&5&0 \end{array}} \right]\) , then which of the following is/are correct?
1. |AB| = ± 47
2. AB = BA
Select the correct answer using the code given below:If \({\rm{E}}\left( {\rm{\theta }} \right) = \left[ {\begin{array}{*{20}{c}} {\cos {\rm{\theta }}}&{\sin {\rm{\theta }}}\\ { - \sin {\rm{\theta \;}}}&{\cos {\rm{\theta }}} \end{array}} \right]\) then E(α) E(β) is equal to
