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Higher Study | Engineering Mathematics | Linear Algebra

Which one of the following equations is a correct identity for arbitrary 3 × 3 real matrices P, Q and R?
Consider a 3 × 3 matrix ? whose (i, j)-th element, ai, j = (i - j)3 Then the matrix A will be
Consider a transformation T: R3 → R2 where R3 and R2 represent three and two-dimensional real column vectors respectively. Also, T(x) = Ax for some matrix A and for each x in R3. How many rows and columns do A have and what is it's maximum possible rank?
A set of linear equations is given in the form Ax = b, where A is a 2 × 4 matrix with real number entries and b ≠ 0. Will it be possible to solve for x and obtain a unique solution by multiplying both left and right sides of the equation by AT (the super script T denotes the transpose) and inverting the matrix AT A? Answer is 
Let A be an n × n matrix with rank r(0 < r < n). Then Ax = 0 has p independent solutions, where p is
Consider the following assertions: I. Rank (ST) = Rank S = Rank T II. Rank (ST) = Rank S, if T is non-singular III. Rank (ST) = Rank T, if T is non-singular Where S, T : V → V are linear transformations of a finite dimensional vector space V. Which of these is/are correct?
Eigen values of a 4 × 4 matrix [A] are given as 2, -3, 13 and 7. Then, what is the value of |det[A]| ?
What is the value of \(\left| {\begin{array}{*{20}{c}} {{b^2} + {c^2}}&{ab}&{ac}\\ {ba}&{{c^2} + {a^2}}&{bc}\\ {ca}&{cb}&{{a^2} + {b^2}} \end{array}} \right|\)
What could be the rank R of a matrix Am × n, if m < n, and A is not a null matrix?
The difference between the digit of a two-digit number is 4. What is the digit in unit’s place? To find out the answer, which of the information given in the statement P and Q is/are sufficient. P: The difference between the actual number and the number obtained by interchanging the positions of the digit is 36. Q: The sum of the digits of the number is 12.
In the matrix equation Px = q, which of the following is a necessary condition for existence of at least one solution for unknown vector x?
A is an (m × n) matrix with m > n and ‘I’ is Identity matrix. Let A1 = (AT A)-1 AT, Then which of the following statement is false?
If v1, v2, …., v6 are six vectors in R4, which one of the following statements is FALSE?
Consider the systems, each consisting of m linear equations in n variables. I. If m < n, then all such systems have a solution II. If m > n, then none of these systems has a solution III. If m = n, then there exists a system which has a solution Which one of the following is CORRECT? 
For Multiplication of matrix which of the following is true
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