Let \( A \) be a matrix of order \( 3 \times 3 \) and \( |A| = 4 \). If \( |2 adj(3A)| = 2^{\alpha} 3^{\beta} \), then what is the value of \( (\alpha + \beta) \)?
If \( \alpha \) and \( \beta \) are the distinct roots of equation \( x^{2} - x + 1 = 0 \), then what is the value of \( \left|\frac{\alpha^{100} + \beta^{100}}{\alpha^{100} - \beta^{100}}\right| \)?
Let \( A \) and \( B \) be symmetric matrices of same order, then which one of the following is correct regarding \( (AB - BA) \)?
1. Its diagonal entries are equal but nonzero
2. The sum of its non-diagonal entries is zero
Select the correct answer using the code given below:
Consider the following statements in respect of square matrices \( A, B, C \) each of same order \( n \):
1. \( AB = AC \Rightarrow B = C \) if \( A \) is non-singular
2. If \( BX = CX \) for every column matrix \( X \) having \( n \) rows then \( B = C \)
Which of the statements given above is/are correct?
Let \( AX = B \) be a system of 3 linear equations with 3-unknowns. Let \( X_{1} \) and \( X_{2} \) be its two distinct solutions. If the combination \( aX_{1} + bX_{2} \) is a solution of \( AX = B \); where \( a, b \) are real numbers, then which one of the following is correct?
