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.
Similar Questions
Consider the following in respect of matrices A, B and C of same order:
1) (A + B + C)' = A' + B’ + C’
2) (AB)’ = A’B’
3) (ABC)’ = C’B’A’
Where A’ is the transpose of the matrix A.
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A square matrix A is called orthogonal if_______ where A’ is the transpose of A.
Consider the following in respect of matrices A and B of same order:
1) A 2– B 2= (A + B) (A – B)
2) (A – I) (I + A) = O ⇔ A 2= I
Where I is the identity matrix and O is the null matrix.
Which of the above is/are correct?If \({\rm{f}}\left( {\rm{x}} \right) = \left[ {\begin{array}{c} {\cos {\rm{x}}}&{ - \sin {\rm{x}}}&0\\ {\sin {\rm{x}}}&{\cos {\rm{x}}}&0\\ 0&0&1 \end{array}} \right]\) , then which of the following are correct?
1. f(θ) × f(ϕ) = f(θ + ϕ)
2. The value of the determinant of the matrix f(θ) × f(ϕ) is 1
3. The determinant of f(x) is an even function.
Select the correct answer using the code given below:
Consider the following in respect of the matrices:
A = [m n], B = [-n -m] & \( C = \begin{bmatrix} m \\ -m\end{bmatrix} \)
1. CA = CB
2. AC = BC
3. C(A + B) = CA + CB
Which of the above statements is/are correct?
Important Questions from Operations on Matrices
The adjoint of matrix \(\left[ {\begin{array}{*{20}{c}} a&b\\ c&d \end{array}} \right]\)is
Consider the following in respect of matrices A, B and C of same order:
1) (A + B + C)' = A' + B’ + C’
2) (AB)’ = A’B’
3) (ABC)’ = C’B’A’
Where A’ is the transpose of the matrix A.
Which of the above are correct?If \({\rm{A}} = \left[ {\begin{array}{*{20}{c}} 1&0&{ - 2}\\ 2&{ - 3}&4 \end{array}} \right]\) , then the matrix X for which 2X + 3A = 0 holds true is
Find a matrix X such that 2A + B + X = 0 , where
\(A=\begin{bmatrix} -1 & 2 \\\ 3 & 4 \end{bmatrix} \ \text{and} \;\rm B =\ \begin{bmatrix} 3 & -2 \\\ 1 & 5 \end{bmatrix} \ ?\)
The solution of the matrix equation \(\left[ {\begin{array}{*{20}{c}} 2&{ - 1}&3\\ 1&1&1\\ 1&{ - 1}&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x\\ y\\ z \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 9\\ 6\\ 2 \end{array}} \right]\) is:
